Program and Abstract Book - Xiamen Universitymath.xmu.edu.cn/conference/Brochure4.pdf · Xue-Cheng Tai: A fast algorithm ... A method for numerical analysis of a Lotka -Volterra food

International Conference on Mathematical Modeling, Analysis and Computation, Xiamen University, 2012

I

Program and Abstract Book

International Conference on Mathematical Modeling,

Analysis and Computation (ICM2AC)

July 21-25, 2012

Center for Computational and Applied Mathematics

Xiamen University, China

http://ccam.xmu.edu.cn/


II

Contents Background ....................................................................................................................................... 1

Organizing Committee ...................................................................................................................... 1

Accommodation ................................................................................................................................ 1

Travel ................................................................................................................................................ 2

Invited Participants ........................................................................................................................... 3

Program ............................................................................................................................................. 5

Daily Program ................................................................................................................................... 7

Tour Information ............................................................................................................................. 13

Abstract of Invited Lectures ............................................................................................................ 16 Remi Abgrall: High Order Residual Distribution Scheme for the RANS Equations

....................................................................... 16

Christine Bernardi: Finite Element Discretization of Richards model ..... 16

Jerry Bona: Propagation of long-crested water waves ..................... 16

Claudio Canuto:BDDC preconditioners for Continuous and Discontinuous Galerkin

methods using spectral/hp elements with variable local polynomial degree 17

Hongqiu Chen: Analysis on stability of solitary-wave solutions for systems of

nonlinear dispersive equations .......................................... 17

I-Liang Chern: Exploring ground states and excited states for spin-1

Bose-Einstein condensates ............................................... 18

Min Chen: Study of nonlocal viscous dispersive terms .................... 19

Weizhong Dai: New Linearized Finite Difference Scheme for Solving Nonlinear

Schrodinger Equations ................................................... 19

Qiang Du: How to compute saddle point without Hessian ................... 20

Weinan E: Stability of Laminar Shear Flow ............................... 20

Yinnian He: Implicit/Explicit Schemes for the Navier-Stokes Equations ... 20

Jingfang Huang: Mathematical and Numerical Aspects of the Adaptive Fast

Multipole Poisson-Boltzmann Solver ...................................... 21

Weizhong Huang: Conditioning of finite element equations with arbitrary

anisotropic meshes ...................................................... 22

Lili Ju: A Posteriori Error Analysis of Finite Element Methods for Linear

Nonlocal Diffusion and Peridynamic Models ............................... 22

Ming-Jun Lai: Convergence Analysis of a Finite Difference Scheme for the

Gradient Flow associated with the ROF Model ............................. 23

Jinghong Li: Multi-Physics Simulation of Laser Fusion ................... 23

Peijun Li: An Inverse Stochastic Source Scattering Problem .............. 24

Zhilin Li: Adaptive mesh refinement techniques for the immersed interface method

applied to flow problems ................................................ 24

Qun Lin: A Type of Multigrid Method for Eigenvalue Problems ............. 25

Peter Markowich: On Wigner and Bohmian Measures in semi.classical Quantum

Dynamics ................................................................ 25


III

Alain Miranville: Some equations with logarithmic nonlinear terms ....... 25

Richard Pasquetti: Fekete-Gauss TSEM with application to incompressible flows

....................................................................... 26

Dongwoo Sheen: New Aspects of Quadrilateral Nonconforming Finite Elements 26

John Strain: First-order overdetermined systems for elliptic problems ... 27

Xue-Cheng Tai: A fast algorithm for Eulers elastica model using augmented

Lagrangian method ....................................................... 27

Huazhong Tang: Runge-Kutta Discontinuous Galerkin methods with WENO limiters

for the special relativistic hydrodynamics .............................. 28

Tao Tang: A General Moving Mesh Framework for Simulating Multi-Phase Flows

....................................................................... 28

Jacobus van der Vegt: HP-Multigrid as Smoother algorithm for higher order

discontinuous Galerkin discretizations of advection-dominated flows ..... 29

Li-Lian Wang: Fast time-domain computation of wave scattering problems .. 29

Qi Wang: Modeling nematic gels and solutions ............................ 30

Hong Wang: Anomalous diffusion and fast solution methods for fractional

diffusion equations ..................................................... 30

Xiaoping Wang: On contact angle hysteresis on rough surfaces ............ 31

Xiaoming Wang: Numerical Schemes for models of thin film epitaxy ........ 31

Yang Xiang: Continuum Model and Numerical Simulation for Dynamics of Dislocation

Arrays .................................................................. 31

Jianlin Xia: Randomized direct solvers for large discretized PDEs ....... 32

Ziqing Xie: The study of the critical perturbation value for singularly

perturbed semilinear elliptic problems .................................. 32

Jinchao Xu: Single-Grid Multilevel Method ............................... 33

Xuejun Xu: Dirichlet-Neumann Operator and Robin-type Domain Decomposition 33

Xiaofeng Yang: Energy stable numerical schemes and simulations for two phase

complex fluids on the phase field method ................................ 34

Pingwen Zhang: The small Deborah number limit of the Doi-Onsager equation to

the Ericksen-Leslie equation ............................................ 34

Zhimin Zhang: Superconvergence: Unclaimed Territories (Part 2) .......... 35

Aihui Zhou: Symmetry Based Eigenvalue Computations ...................... 35

Abstract of Mini-symposium Talks ................................................................................................. 36 Yongyong Cai: Uniform error estimates of numerical methods for nonlinear

Schrodinger equation with wave operator ................................. 36

Yanlai Chen: Superconvergence Properties of Variable-degree HDG Methods for

Convection-diffusion Equations on Nonconforming Meshes. ................. 36

Yujia Chen & Colin Macdonald: Solving Surface PDEs via the Closest Point Method

....................................................................... 37

Xuanchun Dong: Numerical analysis for nonlinear Klein-Gordon equation in the

nonrelativistic limit regime ............................................ 37

Kui Du: Numerical computation of the electromagnetic scattering from a

two-dimensional open cavity ............................................. 38

Jianlong Han: A method for numerical analysis of a Lotka-Volterra food web model


IV

....................................................................... 38

Xue Jiang: Adaptive PML finite element method for multiple scattering problems

....................................................................... 39

Huiyuan Li: FFTs on the Hexagonal and FCC lattices on GPUs .............. 39

Guixia Lv & Longjun Shen & Shunkai Sun: Development of finite point method for

partial differential equations I ........................................ 40

Guixia Lv & Longjun Shen & Shunkai Sun: Development of finite point method for

partial differential equations II ....................................... 40

Guohui Song: Sampling with Localized and Weakly-localized Frames ........ 41

Ke Shi: Devising superconvergent hybridizable DG methods for Stokes and

elasticity problem ...................................................... 41

Qinglin Tang: Quantized vortex dynamics in complex Ginzburg-landau equation in

bounded domain .......................................................... 42

Hanquan Wang: An adaptive level set method based on two-level uniform meshes

and its application to dislocation dynamics ............................. 42

Xiaoqiang Wang: Asymptotic Analysis of Phase Field Formulations of Bending

Elasticity Models ....................................................... 43

Yangfang Wang: A multi-scale moving boundary mathematical model for cancer

invasion ................................................................ 43

Haijun Wu: Pre-asymptotic error Analysis of CIP-FEM and FEM for Helmholtz

Equation with high Wave Number .......................................... 44

Yan Xu: Negative-Order Norm Estimates for Nonlinear Hyperbolic Conservation

Laws.................................................................... 44

Yang Yang: Superconvergence of discontinuous Galerkin method for linear

hyperbolic equations in one space dimension ............................. 45

Haijun Yu: Numerical Study of the Instability of Incompressible Channel Flows

....................................................................... 45

Hui Zhang:Self-consistent Mean Field Model of Hydrogel and Its Numerical

Simulation .............................................................. 46

Lei Zhang: Consistent Atomistic/Continuum Coupling ...................... 46

Lei Zhang: Noise Drives Sharpening of Expression Boundaries in the Zebrafish

Hindbrain ............................................................... 47

Weiying Zheng: Multiscale eddy current problems ......................... 47

Shengxin Zhu & Rosemary A.Renaut & Andy J.Wathen: Smoothness matching for radial

basis functions ......................................................... 48

Abstract of Contributed Talks/Posters ............................................................................................. 49 Sheng Fang: Loosely coupled parallel computation of an SVD and applications in

optimization and machine learning ....................................... 49

Lizhen Chen: Parallel spectral element method using direction splitting for the

incompressible Navier-Stokes equations ................................. 49

Hongying Huang: A Coupling of Local Discontinuous Galerkin and Natural Boundary

Element Method for Exterior Problems .................................... 50

Jinyang Huang: A stable algorithm for non-homogeneous waveguide equation based

on DtN maps ............................................................. 50


V

Pengtao Sun: Full Eulerian modeling and effective numerical studies for the

dynamic fluid-structure interaction problem ............................. 51

Qi Wang: Numerical stability of split-step forward Euler method for stochastic

delay differential equations ............................................ 51

Zhiguo Xu: The dynamics of quantized vortices in Ginzburg-Landau Equation based

on particle interaction ................................................. 52

Wenjun Ying: A Fast Accurate Boundary Integral Method for Potentials on Closely

Packed Cells ............................................................ 52

Xuying Zhao: A convergent adaptive finite element algorithm for nonlocal

diffusion models ........................................................ 53

Xiang Zhou: Cross entropy method for multiple modes in rare-event simulation

....................................................................... 53

工作人员及联系电话 ..................................................................................................................... 54


1

Background

This conference is organized by the Center for Computational and Applied

Mathematics of Xiamen University, and is aimed at bringing together leading

researchers in computational and applied mathematics to discuss recent advances on

modeling, analysis and computation of emerging and challenging problems with

applications.

The conference is sponsored by School of Mathematical Sciences, Xiamen University.

Organizing Committee

• Weizhu Bao National University of Singapore • Claude-Michel Brauner Xiamen University • Jianxian Qiu Xiamen University • Jie Shen (Chair) Purdue University, Xiamen University • Zhong Tan Xiamen University • Chuanju Xu Xiamen University

Accommodation

All invited speakers will stay at YiFuLou (Xiamen University International Academic

Exchange Center). There are three recommended hotels for all participants which can

be booked at discounted rates：Keli Building (克立楼), which is located inside the

university by the man-made lake named "Furong Lake", Hilford Hotel (厦门希尔福

酒店), which is ten minutes walk from Xiamen University, and Peony Wanpeng

Hotel.

Keli Building (克立楼):

Types of rooms: One Queen Bed, 270RMB, Two Double Beds, 270RMB, Family

room, 458RMB Tel: (+86) 0592-2182623

Hilford Hotel (厦门希尔福酒店)




http://baike.baidu.com/view/2742183.htm


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Types of rooms: One Queen Bed, 330RMB, Two Double Beds, 330RMB, Tel: (+86)

0592-2082222

Xiamen Peony Wanpeng Hotel (厦门牡丹万鹏酒店)

Types of rooms: One Queen Bed, 380RMB, Two Double Beds, 380RMB, Tel: (+86)

0592-2662888

If you have paid the registration fee and would like us to book a room for you, please

email [email protected] with your choice of hotels (first choice and second choice),

room type and dates. Please note that there is no guarantee for the success of your

choice due to the limited availability of the hotels.

The special conference price is valid until June 20, 2012.

Travel

How to get to YiFuLou (逸夫楼) from the airport

Option 1: By pickup bus at specified time on July 21: We plan to have pickup bus departing from the Gate 12 on the Second-Floor (the departure floor) at 14:00, 16:00, 17:00, 18:00, 19:00, 20:00 and 22:00 (subject to change).

The bus will take you to YiFuLou (逸夫楼)for registration. For those

staying at other hotels, it is only a short distance from YiFuLou. Option 2: By taxi: The cost from the airport to the hotel is about 60 RMB (there may be an additional 20% charge after 23:00). For those who do not speak Chinese, please print out the following and hand it to the taxi driver: 请将我送到：厦门大学学术交流中心（厦门大学逸夫楼）地址：厦门市思明南路 422 号电话：Tel:(+86)0592-2182623

http://www.peony-hotel.com/wanpeng/


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Invited Participants

Remi Abgrall, INRIA, France Christine Bernardi, Universite Pierre et Marie Curie, France Jerry Bona, University of Chicago, USA Claudio Canuto, Politecnico di Torino, Italy Hongqiu Chen, Memphis, USA I-Liang Chen, National Taiwan University Min Chen, Purdue University, USA Weizhong Dai, Louisiana Tech, USA Qiang Du, Penn State University, USA Weinan E, Peking University and Princeton University Yinnian He, Xi'an Jiaotong University, China Jingfang Huang, University of North Carolina, USA Weizhang Huang, University of Kansas, USA Lili Ju, University of south Carolina Ming-Jun Lai, University of Georgia, USA Peijun Li, Purdue University, USA Jinghong Li, Institute of Applied Physics an Computational Mathematics Zhilin Li, NC State University, USA Qun Lin, Chinese Academy of Sciences, China Yanping Lin, HK polytechnic University, Hongkong Peter Markowich, King Abdullah University of Science and Technology, UK Alain Miranville, University of Poitiers, France Richard Pasquetti, University of Nice, France Dongwoo Sheen, Seoul National University, Korea John Strain, UC Berkerley, USA Xuecheng Tai, University of Bergen, Norway Huazhong Tang, Peking University, China Tao Tang, Hong Kong Baptist University Jaap van der Vegt, University of Twente, the Netherlands Junping Wang, USA National Science Foundation, USA Li-Lian Wang, Nanyang Technological University, Singapore Qi Wang, University of South Carolina, USA Hong Wang, University of south Carolina Xiaoming Wang, Florida State University, USA Yang Xiang, Hong Kong University of Science and Technology, China Jianlin Xia, Purdue University Ziqing Xie, Hunan Normal University, China Jinchao Xu, Penn State University, USA Xuejun Xu, Chinese Academy of Science , China Xiaofeng Yang, University of South Carolina, USA Pingwen Zhang, Peking University, China


4

Zhimin Zhang, Wayne State University, USA Aihui Zhou, Chinese Academy of Sciences, China


5

Program Conference Venue： Lecture Room 1,4 and 5 at the Sciences and Art Center

of Xiamen University

University Map


6

Program

July. 21 July. 22 July.23 July.24 July.25 Times Saturday Sunday Monday Tuesday Wednesday 8:30

Registration (12:00-21:00)

Opening remarks Group photo

W.N. E J. Strain

R.Abgrall Lili Ju

J.V.D.Vegt

X.J. Xu L.L.Wang

J.H. Li

J.C. Xu C.Canuto T. Tang

Z.M.Zhang P.J. Li

Y.N.He

L. Zhang (CUHK) H.Y. Li H.Zhang H.J. Yu J.L.Han

10:00 Tea Break Tea Break Tea Break Tea Break

10:30

J. Bona P.W. Zhang

Q. Lin

A.Zhou X.F.Yang

D.W.Sheen

Y.J. Chen G.X. Lv L. Zhang (Oxford)

Y.F.Wang P. T.Sun

P.Markowich

Q. Du

H.Z. Tang H.Wang Y.P. Lin

Y.Y. Cai X.C.Dong Q.L.Tang H.Q.Wang

12:30 Lunch Lunch Lunch Lunch

14:00

I-L. Chen Q.Wang (USA)

Z.L. Li

X. Yang M. Chen

H.Q. Chen

M.J. Lai X.C. Tai Z.Q. Xie

Y. Xu G.H.Song Y.L.Chen Y. Yang K. Shi

Tour

S.X. Zhu X.Y.Zhao W.J. Yin L.Z.Chen

S.Fang Q.Wang(China)

15:45 Tea Break Tea Break Tea Break

16:15 R.Pasquetti C.Bernardi W.Z.Huang

X.M.Wang A.Miranville

X.P.Wang J.F. Huang J.L. Xia W.Z. Dai

H.J. Wu W.Y.Zheng K. Du X.Jiang X.Q.Wang

Z.G.Xu X.Zhou

J.Y.Huang H.Y.Huang

18:00 Dinner Reception Dinner Dinner Banquet


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Daily Program

Conference Venue: Lecture Room 1,4 and 5, Xiamen University Sciences and Art Center

Saturday, July 21, 2012

13:00-22:30 Registration (Venue: Lobby of YiFuLou)

Sunday, July 22, 2012 8:00-12:00 Registration (Venue: Lobby of YiFuLou) 8:20-8:35 Opening Remarks

Room 1 Plenary Session: Chair Jie Shen 8:35-9:15 Stability of Laminar Shear Flow Weinan E

9:15-9:55 First-order overdetermined systems for elliptic problems John Strain

9:55-10:30 Photo, poster session and tea break Room 1 Plenary Session: Chair Remi Abgrall

10:30-11:10 Propagation of long-crested water waves Jerry Bona

11:10-11:50 The small Deborah number limit of the Doi-Onsager equation to the Ericksen-Leslie equation Pingwen Zhang

11:50-12:30 A Type of Multigrid Method for Eigenvalue Problems Qun Lin

12:30-14:00 Lunch break

Room 4 Parallel Session: Chair Xiaoping Wang

14:00-14:35 Exploring ground states and excited states for spin-1 Bose-Einstein condensates I-Liang Chen

14:35-15:10 Modeling nematic gels and solutions Qi Wang(USA）

15:10-15:45 Adaptive mesh refinement techniques for the immersed interface method applied to flow problems Zhilin Li

Room 5 Parallel Session: Chair Zhong Tan

14:00-14:35 Continuum Model and Numerical Simulation for Dynamics of Dislocation Arrays Xiang Yang

14:35-15:10 Study of nonlocal viscous dispersive terms Min Chen

15:10-15:45 Analysis on stability of solitary-wave solutions for systems of nonlinear dispersive equations Hongqiu Chen

15:45-16:15 Tea break

Room 4 Parallel Session: Chair Weizhu Bao


8

16:15-16:50 Fekete-Gauss TSEM with application to incompressible flows

Richard Pasquetti

16:50-17:25 Finite Element Discretization of Richards model Christine Bernardi

17:25-18:00 Conditioning of finite element equations with arbitrary anisotropic meshes

Weizhang Huang

Room 5 Parallel Session: Chair Claude Brauner

16:15-16:50 Numerical schemes for models of thin film epitaxy Xiaoming Wang

16:50-17:25 Some equations with logarithmic nonlinear terms Alan Miranville

17:25-18:00 On contact angle hysteresis on rough surfaces Xiaoping Wang

Monday, July 23, 2012 Room 4 Parallel Session: Chair Jianxian Qiu

8:30-9:05 High Order Residual Distribution Scheme for the RANS Equations Remi Abgrall

9:05-9:40 A Posteriori Error Analysis of Finite Element Methods for Linear Nonlocal Diffusion and Peridynamic Models Lili Ju

9:40-10:15 HP-multigrid as smoother algorithm for higher order discontinuous Galerkin discretizations of advection-dominated flows

J. van der Vegt

Room 5 Parallel Session: Chair I-Liang Chen

8:30-9:05 Dirichlet-Neumann operators and optimized Schwarz methods Xuejun Xu

9:05-9:40 Fast time-domain computation of wave scattering problems

Lilian Wang

9:40-10:15 Multi-Physics Simulation of Laser Fusion Jinghong Li

10:15-10:45 Tea break Room 4 Parallel Session: Chair Qi Wang(USA) 10:45-11:20 Symmetry Based Eigenvalue Computations Aihui Zhou

11:20-11:55 Energy stable numerical schemes and simulations for two phase complex fluids on the phase field method Xiaofeng Yang

11:55-12:30 New Aspects of Quadrilateral Nonconforming Finite Elements

Dongwoo Sheen

Room 5 Mini-Symposium: Chair Shengxin Zhu


9

10:45-11:05 Solving Surface PDEs via the Closest Point Method Yujia Chen

11:05-11:25 Development of finite point method for partial differential equations Guixia Lv

11:25-11:45 Consistent Atomistic/Continuum Coupling Lei Zhang (Oxford)

11:45-12:05 A multi-scale moving boundary mathematical model for cancer invasion Yangfang Wang

12:05-12:25 Full Eulerian modeling and effective numerical studies for the dynamic fluid-structure interaction problem Pengtao Sun

Room 4 Parallel Session: Chair Aihui Zhou

14:00-14:35 Convergence Analysis of a Finite Difference Scheme for the Gradient Flow associated with the ROF Model Ming-jun Lai

14:35-15:10 A fast algorithm for Eulers elastica model using augmented Lagrangian method Xuecheng Tai

15:10-15:45 The study of the critical perturbation value for singularly perturbed semilinear elliptic problems Ziqing Xie

Room 5 Mini-Symposium: Chair Yanlai Chen

14:00-14:25 Negative-Order Norm Estimates for Nonlinear Hyperbolic Conservation Laws Yan Xu

14:25-14:50 Sampling with Localized and Weakly-localized Frames Guohui Song

14:50-15:15 Superconvergence Properties of Variable-degree HDG Methods for Convection-diffusion Equations on Nonconforming Meshes

Yanlai Chen

15:15-15:40 Superconvergence of discontinuous Galerkin method for linear hyperbolic equations in one space dimension Yang Yang

15:40-16:05 Devising superconvergent hybridizable DG methods for Stokes and elasticity problem Ke Shi

15:45-16:15 Tea break Room 4 Parallel Session: Chair Xuecheng Tai

16:15-16:50 Mathematical and Numerical Aspects of the Adaptive Fast Multipole Poisson-Boltzmann Solver Jingfang Huang

16:50-17:25 Randomized direct solvers for large discretized PDEs Jianlin Xia

17:25-18:00 A New Linearized Finite Difference Scheme for Solving Nonlinear Schrodinger Equations Weizhong Dai

Room 5 Mini-Symposium: Chair Peijun Li


10

16:15-16:40 Pre-asymptotic error Analysis of CIP-FEM and FEM for Helmholtz Equation with high Wave Number Haijun Wu

16:40-17:05 Multiscale eddy current problems Weiying Zheng

17:05-17:30 Numerical computation of the electromagnetic scattering from a two-dimensional open cavity Kui Du

17:30-17:55 Adaptive PML finite element method for multiple scattering problems Xue Jiang

17:55-18:20 Asymptotic Analysis of Phase Field Formulations of Bending Elasticity Models

Xiaoqiang Wang

Tuesday, July 24, 2012

Room 1 Plenary Session: Chair Chuanju Xu 8:30-9:10 Single-Grid Multilevel Method Jinchao Xu

9:10-9:50 BDDC preconditioners for Continuous and Discontinuous Galerkin methods using spectral/hp elements with variable local polynomial degree

Claudio Canuto

9:50-10:30 A General Moving Mesh Framework for Simulating Multi-Phase Flows Tao Tang

10:30-11:10 Poster session and tea break Room 1 Plenary Session: Chair J. van der Vegt

11:10-11:50 On Wigner and Bohmian Measures in semi-classical Quantum Dynamics

Peter Markowich

11:50-12:30 How to compute saddle point without Hessian Qiang Du 12:30-13:30 Lunch break 13:30-18:00 Tour 18:30--21:00 Banquet at Xiamen Hotel

Wednesday, July 25, 2012 Room 4 Parallel Session: Chair Huazhong Tang 8:30-9:05 Superconvergence: Unclaimed Territory, Part 2 Zhiming Zhang

9:05-9:40 An Inverse Stochastic Source Scattering Problem Peijun Li

9:40-10:15 Implicit/Explicit Schemes for the Navier-Stokes Equations Yinnian He

Room 5 Mini-Symposium: Chair Xiaofeng Yang

8:30-8:55 Noise Drives Sharpening of Expression Boundaries in the Zebrafish Hindbrain

Lei Zhang (CUHK)


11

8:55-9:20 FFTs on the Hexagonal and FCC lattices on GPUs Huiyuan Li

9:20-9:45 Self-consistent Mean Field Model of Hydrogel and Its Numerical Simulation Hui Zhang

9:45-10:10 Numerical Study of the Instability of Incompressible Channel Flows Haijun Yu

10:10-10:35 A method for numerical analysis of a Lotka-Volterra food web model Jianlong Han

10:15-10:45 Tea break Room 4 Parallel Session: Chair Zhimin Zhang

10:45-11:20 Runge-Kutta Discontinuous Galerkin methods with WENO limiters for the special relativistic hydrodynamics

Huazhong Tang

11:20-11:55 Anomalous diffusion and fast solution methods for fractional diffusion equations Hong Wang

11:55-12:30 TBA Yanping Lin Room 5 Mini-Symposium: Chair Yongyong Cai

10:45-11:10 Uniform error estimates of numerical methods for nonlinear Schrodinger equation with wave operator Yongyong Cai

11:10-11:35 Numerical analysis for nonlinear Klein-Gordon equation in the nonrelativistic limit regime Xuanchun Dong

11:35-12:00 Quantized vortex dynamics in complex Ginzburg-Landau equation in bounded domain Qinglin Tang

12:00-12:25 An adaptive level set method based on two-level uniform meshes and its application to dislocation dynamics

Hanquan Wang

12:30-13:20 Lunch break Room 5 Contributed Talks: Chair Hongtao Chen

13:30-13:50 Smoothness matching for radial basis functions Shengxin Zhu

13:50-14:10 A convergent adaptive finite element algorithm for nonlocal diffusion models Xuying Zhao

14:10-14:30 A Fast Accurate Boundary Integral Method for Potentials on Closely Packed Cells Wenjun Yin

14:30-14:50 Parallel spectral element method using direction splitting for the incompressible Navier-Stokes equations

Lizhen Chen


12

14:50-15:10 Loosely coupled parallel computation of an SVD and applications in optimization and machine learning Sheng Fang

15:10-15:30 Numerical stability of split-step forward Euler method for stochastic delay differential equations Qi Wang(China)

15:30-15:50 Tea Break Room 5 Contributed Talks: Chair Kui Du

15:50-16:10 The dynamics of quantized vortices in Ginzburg-Landau Equation based on particle interaction

Zhiguo Xu

16:10-16:30 Cross entropy method for multiple modes in rare-event simulation Xiang Zhou

16:30-16:50 A stable algorithm for non-homogeneous waveguide equation based on DtN maps Jinyang Huang

16:50-17:10 A Coupling of Local Discontinuous Galerkin and Natural Boundary Element Method for Exterior Problems

Yongying Huang


13

Tour Information

July 24, afternoon: a tour to Gulangyu (鼓浪屿)

As a place of residence for Westerners during Xiamen's colonial past, Gulangyu is famous for its architecture and for hosting China's only piano museum, giving it the nickname of "Piano Island" or "The Town of Pianos" or "The Island of Music". There are over 200 pianos on this island. The Chinese name also has musical roots, as Gu lang means drum waves so-called because of the sound generated by the ocean waves hitting the reefs. Yu means "islet". In addition, there is a museum dedicated to Koxinga, Haidi Shijie Marine World, a subtropical garden containing plants introduced by overseas Chinese, as well as Xiamen Museum, formerly the Eight Diagrams Tower. There's also an Organ museum, bird sanctuary, plant nursery, and a tram that takes to the peak. On the west beach of the island you can rent pedal boats and jet skis. There's a garden of 12 grottos to represent each of the animals on the zodiac. Built into the hillside, its a maze of caves and tunnels to find all twelve (and the exit). There are many boutique hotels to stay in as well. The island of Gulangyu is a pedestrian only destination, where the only vehicles on the islands are several fire trucks and electric tourist buggies. The narrow streets on the island, together with the architecture of various styles around the world, give the island a unique appearance.

July 26: one day tour to Nanjing Tulou (南靖土楼一日游 260 RMB)

Expected to Return to Airport/Hotel before 17:30

http://en.wikipedia.org/wiki/Piano

http://en.wikipedia.org/wiki/Reef

http://en.wikipedia.org/wiki/Islet

http://en.wikipedia.org/wiki/Koxinga

http://en.wikipedia.org/wiki/Subtropical

http://en.wikipedia.org/wiki/Plant

http://en.wikipedia.org/wiki/Overseas_Chinese


14

Fujian Tulou is a type of Chinese rural dwellings of the Hakka and Minnan people in the mountainous areas in southeastern Fujian, China. They were mostly built between the 12th and the 20th centuries. A tulou is usually a large, enclosed and fortified earth building, most commonly rectangular or circular in configuration, with very thick load-bearing rammed earth walls between three and five stories high and housing up to 80 families. Smaller interior buildings are often enclosed by these huge peripheral walls which can contain halls, storehouses, wells and living areas, the whole structure resembling a small fortified city. The fortified outer structures are formed by compacting earth, mixed with stone, bamboo, wood and other readily available materials, to form walls up to 6 feet (1.8 m) thick. Branches, strips of wood and bamboo chips are often laid in the wall as additional reinforcement. The result is a well-lit, well-ventilated, windproof and earthquake-proof building that is warm in winter and cool in summer. Tulous usually have only one main gate, guarded by 4–5-inch-thick (100–130 mm) wooden doors reinforced with an outer shell of iron plate. The top level of these earth buildings has gun holes for defensive purposes. A total of 46 Fujian Tulou sites, including Chuxi tulou cluster, Tianluokeng tulou cluster, Hekeng tulou cluster, Gaobei tulou cluster, Dadi tulou cluster, Hongkeng tulou cluster, Yangxian lou, Huiyuan lou, Zhengfu lou and Hegui lou, have been inscribed in 2008 by UNESCO as World Heritage Site, as "exceptional examples of a building tradition and function exemplifying a particular type of communal living and defensive organization in a harmonious relationship with their environment".

Xiamen

Xiamen is a major city on the southeast (Taiwan Strait) coast of the People's Republic of China. It is administered as a sub-provincial city of Fujian province with an area of 1575.16 km2 and population of 3.61 million. Its built up area is now bigger than the old urban island area and covers now all six districts of Xiamen (Huli, Siming, Jimei, Tong'an, Haicang and recently Xiang'an), for a total of 3,531,147 inhabitants. It borders Quanzhou to the north and Zhangzhou making with this city a unique built up area of more than 5 million people. The Jinmen (Kinmen) Islands administered by the Republic of China (Taiwan) are less than 10 kilometers (6.2 mi) away. Xiamen and the surrounding southern Fujian countryside are the ancestral home to large communities of overseas Chinese in Southeast Asia and Taiwan. The city was a treaty port in the 19th century and one of the four original Special Economic Zones opened to foreign investment and trade when China began economic reforms in the early 1980s. It is endowed with educational and cultural institutions supported by the overseas Chinese diaspora. In 2006, Xiamen was ranked as China's second "most suitable city for living" as well as China's "most romantic leisure city" in 2011.

http://en.wikipedia.org/wiki/Hakka_people

http://en.wikipedia.org/wiki/Minnan

http://en.wikipedia.org/wiki/Fujian

http://en.wikipedia.org/wiki/Rammed_earth

http://en.wikipedia.org/wiki/Tianluokeng_tulou_cluster



http://en.wikipedia.org/wiki/UNESCO

http://en.wikipedia.org/wiki/World_Heritage_Site

http://en.wikipedia.org/wiki/Taiwan_Strait

http://en.wikipedia.org/wiki/People%27s_Republic_of_China



http://en.wikipedia.org/wiki/Sub-provincial_city


http://en.wikipedia.org/wiki/Huli_District

http://en.wikipedia.org/wiki/Siming

http://en.wikipedia.org/wiki/Jimei

http://en.wikipedia.org/wiki/Tong%27an

http://en.wikipedia.org/wiki/Haicang_District

http://en.wikipedia.org/wiki/Xiang%27an

http://en.wikipedia.org/wiki/Quanzhou

http://en.wikipedia.org/wiki/Zhangzhou

http://en.wikipedia.org/wiki/Jinmen

http://en.wikipedia.org/wiki/Republic_of_China

http://en.wikipedia.org/wiki/Taiwan


http://en.wikipedia.org/wiki/Southeast_Asia


http://en.wikipedia.org/wiki/Treaty_ports

http://en.wikipedia.org/wiki/Special_Economic_Zones_of_the_People%27s_Republic_of_China


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Xiamen and its surrounding countryside is known for its scenery and tree-lined beaches. Gulangyu, also known as Piano Island, is a popular weekend getaway with views of the city and features many Victorian-era style European edifices. Xiamen's Botanical Garden is a nature lover's paradise. The Buddhist Nanputuo Temple, dating back to the Tang Dynasty, is a national treasure. Xiamen is also famous for its history as a frontline in the Chinese Civil War with Taiwan over Jinmen (also known as Jinmen or Quemoy) 50 years ago. One attraction for tourists is to view Kinmen, a group of islands a few kilometers away and under Taiwanese control, from Xiamen island. Xiamen University. Beautiful campus with old traditional buildings and a tranquil lake outside the foreign language department. Nowadays the University is open for anyone, some registration may needed on peak holidays. To avoid long registration procedure, you may enter the university from three smaller gates which are across the Baicheng beach. Water Garden Expo Park. Water Garden Expo Park has a planning area of about 6.76 square kilometers (land area of 3.03 square kilometers), which consists of five exhibition park islands, four ecological landscapes islands and two peninsulas, including the main pavilion, Chinese Education Park, Marine Culture Island, Spa Island and other functional areas and related facilities

Xiamen University

Xiamen University colloquially known as Xia Da, located in Xiamen, Fujian province, is the first university in China founded by overseas Chinese. Before 1949, it was originally known as the University of Amoy. The school motto is "Pursue Excellence, Strive for Perfection". The university is one of many comprehensive universities directly administered by the Chinese Ministry of Education. In 1995 it was included in the list of the 211 Project for the state key construction; in 2000 it became one of China’s higher-level universities designated for the state key construction of the 985 Project. According to University Undergraduates Teaching Assessment and Chinese Universities Evaluation Standings, the university is ranked 11th in China, which is outside the top ten and has maintained the top 20 ranking in China, among which 6 subjects reach A++ level, including economics and management, music, law, chemistry, journalism, communication and mathematics.

http://en.wikipedia.org/wiki/Gulangyu_Island

http://en.wikipedia.org/wiki/Nanputuo_Temple

http://en.wikipedia.org/wiki/Chinese_Civil_War


http://en.wikipedia.org/wiki/Jinmen

http://en.wikipedia.org/wiki/Kinmen

http://en.wikipedia.org/wiki/Xiamen_University

http://en.wikipedia.org/w/index.php?title=Water_Garden_Expo_Park&action=edit&redlink=1

http://en.wikipedia.org/wiki/Xiamen



http://en.wikipedia.org/wiki/211_Project





16

Abstract of Invited Lectures

High Order Residual Distribution Scheme for the RANS

Equations

Remi Abgrall INRIA, France

([email protected])

Abstract In this work the use of Residual Distribution schemes for the discretization of conservation laws is described. The scheme uses a Lagrangian interpolation of the solution over the element to achieve high order spatial accuracy for steady problems. The formulation is compact and easily generalizable to all spatial dimensions and to all type of elements. The accuracy of the numerical scheme is proved by numerical experiments on benchmark problems in two and three spatial dimensions. Furthermore, the results for the Reynolds Averaged Navier-Stokes computation, based on the Spalart-Allmaras model, of compressible turbulent flows are reported.

Finite Element Discretization of Richards model

Christine Bernardi Universite Pierre et Marie Curie, France

Abstract

We consider the equation due to Richards which models the water flow in a partially saturated underground porous medium under the surface. We propose a discretization of this equation by an implicit Euler’s scheme in time and mixed finite elements in space. We perform the a posteriori analysis of this discretization, in order to improve its efficiency via time step and mesh adaptivity. Some numerical experiments confirm the interest of this approach.

Propagation of long-crested water waves

Jerry Bona University of Illinois at Chicago, USA

([email protected] )

Abstract The lecture will be concerned with water waves that primarily propagate in one direction, but

mailto:[email protected]



17

where allowance is made for weak variation in the direction orthogonal to that of the primary motion. Both waves localized in the direction of primary propagation and bore-like waves will be considered. _____________________________________________________________________

BDDC preconditioners for Continuous and Discontinuous

Galerkin methods using spectral/hp elements with

variable local polynomial degree

Claudio Canuto Politecnico di Torino, Torino, ITALY


Abstract Locally adapted meshes and polynomial degrees can greatly improve spectral element accuracy and applicability. A Balancing Domain Decomposition by Constraints (BDDC) preconditioner is constructed and analyzed for both continuous and discontinuous Galerkin discretizations of scalar elliptic problems, built by nodal spectral elements with variable polynomial degrees. The discontinuous Galerkin case is reduced to the continuous case via the Auxiliary Space Method. The proposed BDDC preconditioner is proven to be scalable in the number of subdomains and quasioptimal in both the ratio of local polynomial degrees and element sizes and the ratio of subdomain and element sizes. Several numerical experiments in the plane con firm the obtained theoretical convergence rate estimates, and illustrate the preconditioner performance for both continuous and discontinuous Galerkin discretizations. Different con figurations with locally adapted polynomial degrees are studied, as well as the preconditioner robustness with respect to discontinuities of the elliptic coefficients across subdomain boundaries. These results apply also to other dual-primal preconditioners de fined by the same set of primal constraints, such as FETI-DP preconditioners.

Analysis on stability of solitary-wave solutions for

systems of nonlinear dispersive equations

Hongqiu Chen University of Memphis

Abstract

Considered here is a class of systems

(0.1)

of coupled KdV-equations introduced by Bona, Cohen and Wang, where u = u(x,t),v = v(x,t) are



18

functions defined on ,R R+× 2 2( , )P u v Au Buv Cv= + + and 2 2( , )Q u v Du Euv Fv= + + in

which A,B,··· ,F are real numbers. Moreover, if the system of linear equations

(0.2)

has solutions (a,b,c) such that 24ac b> , then, there are up-to three explicit solitary-wave

solutions of hyperbolic square functions. The stability and instability of a solitary-wave solution can be checked by a simple and straightforward algebraic relation.

_____________________________________________________________________

Exploring ground states and excited states for spin-1

Bose-Einstein condensates

I-Liang Chern National Taiwan University ([email protected])

Abstract

In this talk, I will first give a brief introduction to the spinor Bose-Einstein condensates. Then I will present two recent results, one is numerical, the other is analytical. In 1925, Bose and Einstein predicted that massive bosons could occupy the same lowest-energy state at low temperature and formed the so-called Bose-Einstein condensates (BECs). It was realized on several alkali atomic gases in 1995 by laser cooling technique. In the numerical study of spinor BEC, a pseudo-arclength continuation method (PACM) was proposed and employed to compute the ground state and excited state solutions of spin-1 BEC. Numerical results on the wave functions and their corresponding energies of spin-1 BEC with repulsive/attractive and ferromagnetic/antiferromagnetic interactions are presented. Furthermore, it is found that the component separation and population transfer between the different hyperne states can only occur in excited states due to the spin-exchange interactions. In the analytical study, the ground states of spin-1 BEC are characterized. For ferromagnetic systems, we show the validity of the so-called single-mode approximation (SMA). For antiferromagnetic systems, there are two subcases. When the total magnetization M 6= 0, the corresponding ground states have vanishing zeroth (mF = 0) components, thus are reduced to two-component systems. When M = 0, the ground states are also reduced to the SMA, and there are one-parameter families of such ground states. The key idea is a redistribution of masses among different components, which reduces kinetic energy in all situations, and makes our proofs simple and unified. Finally, a fast algorithm based on the above analytic result is provided. It is shown that the new method is about 10 to 20 faster than an old method of Bao-Lim's continuous normalized gradient method. The numerical part is a joint work with Jen-Hao Chen and Weichung Wang, the fast algorithm part is with Weizhu Bao and Yanzhi Zhang, whereas the analytical part is jointly with Liren Lin.



19

Study of nonlocal viscous dispersive terms

Min Chen Purdue University, USA ([email protected])

Abstract

In this talk, we investigate water wave models with nonlocal viscous terms in time or in space, namely

0

/2 1 1/2

( )

( | | sgn( ) ( )) 0

t tt x xxx x xx

ut txx x

u su u u ds uu ut s

u u D F i u uu

νβ νπ

β ν ψ ψ ψ γ−

+ + + + =

− − + + + =

∫

The talk will be centered at the asymptotic behavior of the solutions. Although these two equations are related and under certain conditions formally equivalent, the challenges encountered in their theoretical and numerical investigations are quite different. .

New Linearized Finite Difference Scheme for Solving

Nonlinear Schrodinger Equations

Weizhong Dai College of Engineering and Science, Louisiana Tech University

([email protected])

Abstract The nonlinear Schrodinger equation has been extensively used in physics research, particularly

in the modeling of nonlinear dispersion waves. There are many numerical methods for solving the nonlinear Schrodinger equation that have been developed in the literature. In this talk, a new linearized finite difference scheme for solving nonlinear Schrodinger equations will be presented. The scheme is obtained based on the generalized finite-difference time-domain (FDTD) method coupled with the Crank-Nicolson method. As such, the obtained scheme satisfies the discrete energy conservation laws and provides accurate solutions. This new method is then tested by simulating the propagation and formation of solitons.

_____________________________________________________________________




20

How to compute saddle point without Hessian

Qiang Du Penn State University, USA

Abstract

Exploring complex energy landscape is a challenging issue in many applications. Besides

locating equilibrium states, it is often also important to identify the transition states given by saddle points. In this talk, we will discuss numerical algorithms, in particular, the shrinking dimer dynamics, for the computation of such transition states and present some recently developed mathematical theory. We will consider a number of applications including the study of critical nuclei morphology in solid state transformations, optimal photonic crystal design and the generalized Thomson problem.

_____________________________________________________________________

Stability of Laminar Shear Flow

Weinan E Peking University and Princeton University

Abstract In 1883, Reynolds published his classical work on the experimental study of the stability of shear

shows. Since then the issue of the critical Reynolds number at which laminar flows become unstable has been studied by numerous people, includig Sommerfeld, Heisenberg, C. C. Lin, Orazag, and more recently, Trefethen, Hof, Barkley, Eckhardt, etc. Despite this great deal of effort, the theoretical question as to how the critical Reynolds number should be determined still remains open. In this talk, we present an approach using ideas drawn from statistical physics and large deviation theory. This is joint work with Xiaoliang Wan and Haijun Yu.

_____________________________________________________________________

Implicit/Explicit Schemes for the Navier-Stokes

Equations

Yinnian He School of Mathematics and Statistics, Xi’an Jiaotong University

([email protected])



21

Abstract We consider the implicit/explicit Schemes for the Navier-Stokes Equations and prove the

stabilities and optimal error estimates under the corresponding stability conditions, where the

schemes are almost unconditionally stable and convergent for the smooth initial data 20u H∈ ,

i.e., the time step size τ satisfies 0Cτ < ; and the schemes are almost weak unconditionally stable

and convergent for the non-smooth initial data 10u H∈ , i.e., the time step size τ satisfies

0log h Cτ < for the mesh size 0 < h < 1; and the schemes are conditionally stable for the

non-smooth initial data 20u L∈ , i.e., the time step size τ satisfies 2

0h Cτ − < .

Moreover, the Euler implicit/explicit scheme based on the mixed finite element or called as the time-space finite element iterative method (the T-S method) is applied to solve the stationary Navier-Stokes equations. Finally, the almost unconditionally stability is proven and the optimal error estimates uniform in time are provided for the scheme. Compared the standard the Stokes, Newton and Oseen finite element iterative methods, the T-S Method is an efficiency methods for solving the stationary Navier-Stokes problem with a slightly small viscosity ν and save a large amount of computational time.

Mathematical and Numerical Aspects of the Adaptive Fast

Multipole Poisson-Boltzmann Solver

Jingfang Huang University of North Carolina at Chapel Hill

([email protected])

Abstract This talk summarizes the mathematical and numerical theories and implementation details of the

Adaptive Fast Multipole Poisson-Boltzmann (AFMPB) solver, including the Poisson-Boltzmann model, boundary integral equation reformulation, surface mesh generation, node-patch discretization, Krylov iterative methods, new version of fast multipole methods (FMMs), and dynamic prioritization technique for the spatio-temporal directed acyclic FMM graph on multicore computers. Possible strategies to further improve the efficiency, accuracy and applicability of the AFMPB solver to large-scale long-time molecular dynamics simulations are discussed, and preliminary numerical results are presented to demonstrate the potential of the solver.



22

Conditioning of finite element equations with

arbitrary anisotropic meshes

Weizhang Huang University of Kansas

([email protected])

Abstract This talk is concerned with the conditioning of the linear system resulting from the finite element

discretization of an anisotropic diffusion problem with arbitrary meshes. New bounds on the condition number of the stiffness matrix will be presented. These bounds are shown to depend on three major factors, a factor representing the base order corresponding to the condition number for a uniform mesh, a factor representing the effects of the mesh M-nonuniformity, and a factor representing the effects of the mesh volume-nonuniformity. Diagonal scaling for the finite element linear system and its effects on the conditioning will be addressed. It is shown that a properly chosen diagonal scaling can eliminate the effects of the mesh volume-nonuniformity and reduce the effects of the mesh M-nouniformity on the conditioning of the stiffness matrix. Numerical examples will be presented to verify the theoretical findings.

A Posteriori Error Analysis of Finite Element Methods

for Linear Nonlocal Diffusion and Peridynamic Models

Lili Ju Department of Mathematics, University of South Carolina, USA

([email protected])

Abstract In this talk, we present some results on a posteriori error analysis of finite element methods for solving linear nonlocal diffusion and bond-based peridynamic models. In particular, we aim to propose a general abstract framework for a posteriori error analysis of the peridynamic problems. A posteriori error estimators are consequently prompted, the reliability and efficiency of the estimators are proved. Connections between the a posteriori error estimations of the nonlocal problems and that of the related classical partial differential equation based problems are studied within continuous finite element spaces. Some numerical experiments are also given to test the theoretical conclusions.




23

Convergence Analysis of a Finite Difference Scheme for

the Gradient Flow associated with the ROF Model

Ming-Jun Lai Dept. of Math., University of Georgia

([email protected])

Abstract We present a convergence analysis of a finite difference scheme for the time dependent gradient flow associated with the Rudin-Osher-Fatemi model. We devise an iterative algorithm to compute the solution of the finite difference scheme and prove the convergence of the iterative algorithm. Finally computational experiments are shown to demonstrate the convergence of the finite difference scheme. An application for image denoising is given.

Multi-Physics Simulation of Laser Fusion

Jinghong Li Institute of Applied Physics and Computational Mathematics

([email protected])

Abstract Inertial confinement fusion (ICF) is an approach to fusion that relies on the inertia of the fuel mass to provide confinement [1, 2]. Laser indirect-drive ICF is very complex, and there are many physical processes happened [3]. Multi-physics radiation hydrodynamics codes play a key role in understanding the ICF physics. LARED-Integration, developed under an object-oriented structured-mesh parallel code-supporting infrastructure - JASMIN [4], is a 2-D multi-physics ICF simulation code for studying laser indirect-drive ICF in our institute. The code includes seven main modules: laser light, hydrodynamics, electron, ion, radiation, atomic physics, and burn products. 3-D ray-tracing technique is used for laser light propagation and absorption. Hydrodynamics (2-D) calculation relies on Lagrangian method on quadrilateral grid and Arbitrary Lagrange Eulerian (ALE) method. We use flux-limited thermal conduction for electrons and ions, and use several methods for radiation modeling: thermal conduction, multi-group flux-limited diffusion, multi-group transfer (discrete ordinate SN). Atomic physics includes average atom modeling, Tabulated Opacity (LTE), and Tabulated EOS (LTE). Burn product has two choices: local deposition and Charged particle transfer (to be installed). Here, we present some multi-physics simulations of laser fusion by using multi-group diffusion / transfer radiation methods. References [1] J. D. Lindl, Phys. Plasmas 2, 3933 (1995)



24

[2] J. D. Lindl, et al., Phys. Plasmas 11, 339 (2004) [3] W. B. Pei, Commun. Comput. Phys. 2, 255 (2007) [4] http://www.iapcm.ac.cn/jasmin _____________________________________________________________________

An Inverse Stochastic Source Scattering Problem

Peijun Li Department of Mathematics, Purdue University

([email protected])

Abstract This talk is concerned with an inverse random source scattering problem for the one dimensional stochastic Helmholtz equation in a slab of inhomogeneous medium, where the source function is driven by the Wiener process. Since the source and hence the radiating field are stochastic, the inverse problem is to reconstruct the statistical structure, such as the mean and the variance, of the source function from the measured random field on the boundary point. Based on the constructed solution for the direct problem, integral equations are derived to reconstruct the mean and the variance of the source function. Numerical experiments will be presented to demonstrate the validity and effectiveness of the proposed method.

Adaptive mesh refinement techniques for the immersed

interface method applied to flow problems

Zhilin Li North Carolina State University

([email protected])

Abstract Adaptive mesh refinement strategies are proposed for the Immersed Boundary and Immersed Interface methods for two-dimensional elliptic interface problems, Stokes and Navier-Stokes equations with moving interfaces. The interface is represented by the zero level set of a Lipschitz

function ( , )x yΦ . Our adaptive mesh refinement is done within a small tube of

| ( , ) |x y δΦ < with finer Cartesian meshes. Numerical examples for some benchmark problems

show the efficiency of the grid refinement strategy.



25

A Type of Multigrid Method for Eigenvalue Problems

Qun Lin Academy of Mathematics and Systems Science, Chinese Academy of Sciences

([email protected])

Abstract In this lecture, we give a type of multigrid method for eigenvalue problems by finite element discretization. This method needs the similar computational work as the multigrid method for the associated boundary value problem. _____________________________________________________________________

On Wigner and Bohmian Measures in semi.classical

Quantum Dynamics

Peter Markowich King Abdullah University of Science and Technology, UK

Abstract We present the Schrödinger, Wigner and Bohm approaches to quantum mechanics, discuss connections to fluid dynamics and semiclassical limits, for WKB and coherent initial data.

Some equations with logarithmic nonlinear terms

Alain Miranville Universite de Poitiers, France

([email protected])

Abstract Our aim in this talk is to discuss the well-posedness and asymptotic behavior of several equations with logarithmic nonlinear terms. Such equations arise, e.g., in phase transition.



26

Fekete-Gauss TSEM with application to incompressible

flows

Richard Pasquetti CNRS, France

([email protected])

Abstract Spectral element methods on simplicial meshes, say TSEM, show both the advantages of spectral and finite element methods, i.e., spectral accuracy and geometrical flexibility. We present a TSEM solver of the incompressible Navier-Stokes equations that uses a projection method in time and piecewise polynomial basis functions of arbitrary degree in space. The so-called Fekete-Gauss TSEM is employed, i.e., Fekete (resp. Gauss) points of the triangle are used as interpolation (resp. quadrature) points. For the sake of consistency, isoparametric elements are used to approximate curved geometries. The resolution algorithm is based on an efficient Schur complement method, so that one only solves for the element boundary nodes. Moreover, the algebraic system is never assembled, therefore the number of degrees of freedom is not limiting. An accuracy study is carried out and results are provided for classical benchmarks: the driven cavity flow, the flow between eccentric cylinders and the flow past a cylinder. Details are provided in: L. Lazar, R. Pasquetti, F. Rapetti, Fekete-Gauss spectral elements for incompressible Navier-Stokes flows (2D case), Comm. in Comput. Phys., in press.

New Aspects of Quadrilateral Nonconforming Finite

Elements

Dongwoo Sheen Seoul National University, Korea

([email protected])

Abstract The use of nonconforming elements to solve fluid and fluid mechanical problems has shown several advantages over the use of conforming counterparts. Replacing conforming elements by nonconforming ones is simple and guarantees numerical stability in many cases. We give a brief review on the recent development in quadrilateral nonconforming finite elements. We will then discuss several new interesting observations about these quadrilateral nonconforming elements. Error estimates and numerical results will be presented.



27

First-order overdetermined systems for elliptic

problems

John Strain UC Berkeley Mathematics, USA

([email protected])

Abstract We convert boundary value problems for elliptic partial differential equations to first-order overdetermined systems. The conversion clarifies the analytical connection between solvability of these problems and Protter's classical definition of ellipticity. It suggests efficient alternating direction implicit methods which generalize traditional iterative schemes. It also yields a simple well-conditioned integral equation for arbitrary boundary conditions.

A fast algorithm for Eulers elastica model using

augmented Lagrangian method

Xue-Cheng Tai Univ of Bergen, Norway

([email protected])

Abstract Minimization of functionals related to Euler's elastica energy has a wide range of applications in computer vision and image processing. An issue is that a high order nonlinear partial differential equation (PDE) needs to be solved and the conventional algorithm usually takes high computational cost. In this paper, we propose a fast and efficient numerical algorithm to solve minimization problems related to the Euler's elastica energy and show applications to variational image denoising, image inpainting, and image zooming. We reformulate the minimization problem as a constrained minimization problem, followed by an operator splitting method and relaxation. The proposed constrained minimization problem is solved by using an augmented Lagrangian approach. Numerical tests on real and synthetic cases are supplied to demonstrate the efficiency of our method. Comparisons with the CKS scheme are given. This talk is based on a joint work with J. Hahn and J, Chung.




28

Runge-Kutta Discontinuous Galerkin methods with WENO

limiters for the special relativistic hydrodynamics

Huazhong Tang Peking University

Abstract This paper develops Runge-Kutta discontinuous Galerkin (RKDG) methods with WENO limiters for the one- and two-dimensional special relativistic hydrodynamics, which is an extension of the work [J.X. Qiu and C.-W. Shu, SIAM J. Sci. Comput., 26(2005), 907-929].The key steps of the WENO limiter for the RKDG method are: (1) to identify troubled cells by using a TVB minimod-type limiter; and (2) to reconstruct the polynomial solution inside the “troubled” cells by the WENO reconstruction using the cell averages of neighboring cells, while maintaining the original cell averages of the “troubled” cells. Several numerical examples are given to demonstrate the accuracy and effectiveness of the proposed RKDG methods with WENO limiters.

A General Moving Mesh Framework for Simulating

Multi-Phase Flows

Tao Tang Hong Kong Baptist University

([email protected])

Abstract In this talk, we present an adaptive moving mesh algorithm for meshes of unstructured tetrahedra in three space dimensions. The algorithm automatically adjusts the size of the elements with time and position in the physical domain to resolve the relevant scales in multiscale physical systems while minimizing computational costs. Since the mesh redistribution procedure normally requires to solve large size matrix equations (arising from discretizing the Euler-Lagrange equations or a minimization problem), we will describe a procedure to decouple the matrix equation to a much simpler block-tridiagonal type which can be solved by multi-grid methods efficiently. To demonstrate the performance of the proposed 3D moving mesh strategy, the algorithm is implemented in finite element simulations of fluid-fluid interface interactions in multiphase flows. In this talk, we will propose a general framework on how to design an adaptive grid method useful for this kind of simulations. To demonstrate the main ideas, we consider the formation of drops by using an energetic variational phase field model which describes the motion of mixtures of two incompressible fluids. The phase



29

field model consists of a Navier-Stokes system coupled with volume preserving Allen-Cahn type phase equations. Numerical results on two- and three-dimensional simulations will be presented.

HP-Multigrid as Smoother algorithm for higher order

discontinuous Galerkin discretizations of

advection-dominated flows

Jacobus van der Vegt Department of Applied Mathematics, University of Twente, Netherlands

([email protected])

Abstract Higher order accurate space-time discontinuous Galerkin discretizations are well suited for the solution of time-dependent partial differential equations, in particular for free boundary problems. The space-time DG discretization results, however, in an implicit discretization in time, which requires the solution of a (non)linear algebraic system. In this presentation the new hp-Multigrid as Smoother (hp-MGS) algorithm for higher order accurate DG discretizations of advection-dominated flows will discussed. The main feature of this algorithm is that it uses semi-coarsening h-multigrid as smoother for p-multigrid. An important aspect in the development of the hp-MGS algorithm is a detailed multilevel-level Fourier analysis of the full multigrid algorithm. This analysis gives detailed information on the spectrum and operator norms of the error transformation operator and is used to optimize the multigrid algorithm. The newly derived multigrid algorithm is tested on a various problems, including boundary layers requiring significantly stretched meshes. Joint work with Sander Rhebergen, School of Mathematics, University of Minnesota, Minnesota, USA.

Fast time-domain computation of wave scattering

problems

Li-Lian Wang Division of Mathematical Sciences, School of Physical and Mathematical sciences,

Nanyang Technological University ([email protected])

Abstract




30

Wave propagations in unbounded media arise from diverse applications. Intensive research has been devoted to frequency-domain simulation, e.g., for the time-harmonic Helmholtz problems and Maxwell's equations. Here, we are interested in time-domain computation, which is known to be more exible in capturing wide-band signals and modeling more general material inhomogeneities and nonlinearities. In this talk, we shall show how to use tools in complex analysis to analytically evaluate circular and spherical non-reacting boundary conditions (NRBCs), and how to efficiently deal with time-space globalness of such boundary conditions. Fast spectral-Galerkin solvers together with stable time integration and techniques for handling general irregular scatterers will be introduced for the simulation. We intend to demonstrate that the interplay between analytic tools, accurate numerical means and sometimes brute force hand calculations can lead to efficient methodologies for challenging simulations. This is a joint work with BoWang and Xiaodan Zhao.

Modeling nematic gels and solutions

Qi Wang Department of Mathematics, Interdisciplinary Mathematics Institute NanoCenter at USC

University of South Carolina Columbia ([email protected])

Abstract Nematic gels and solutions are two classes of complex fluids that have had many interesting applications in biology and high performance materials fabrication. In this talk, I will discuss some models developed for the nematic gel and nematic solutions in the context of kinetic theory as well as it continuum description. Active nematics are identified in many novel material systems as instrumental buffer materials to generate power and mechanisms to derive the activity for the material systems. Numerical examples will be provided to illustrate the applicability of the models.

Anomalous diffusion and fast solution methods for

fractional diffusion equations

Hong Wang Department of Mathematics, University of South Carolina

([email protected])

Abstract




31

In this talk we go over anomalous diffusion processes and fractional calculus. We focus on the fast numerical methods for spatial-fractional diffusion equations in multiple space dimensions. Numerical results will be presented to demonstrate the utility of the method.

On contact angle hysteresis on rough surfaces

Xiaoping Wang Hong Kong University of Science and Technology

([email protected])

Abstract We analyze the wetting hysteresis on rough and chemically patterned surfaces from a phase-field model for two phase fluid. In the slow motion, the dynamic of the contact angle can be derived from the matched asymptotic expansions. The contact angle hysteresis is then studied by homogenization as the size of the pattern becomes small.

Numerical Schemes for models of thin film epitaxy

Xiaoming Wang Florida State University, USA

([email protected])

Abstract

We survey recent results on efficient and stable numerical schemes for models of thin film epitaxial growth. We will focus on schemes that preserve the energy law in appropriate discrete form. Both semi-discrete in time as well as fully discretized schemes will be discussed. Numerical results will be presented as well.

Continuum Model and Numerical Simulation for Dynamics

of Dislocation Arrays

Yang Xiang

Hong Kong University of Science and Technology ([email protected] )




32

Abstract We derive a continuum model for the dynamics of dislocation arrays, in which a dislocation array is represented by a continuous surface in three dimensions. The continuum model is derived rigorously from the discrete model of the dynamics of the constituent dislocations in the array using asymptotic analysis. The obtain continuum model contains an integral over the dislocation array surface representing the long-range interaction of dislocations, and a local term that comes from the line tension effect of dislocations. We also present a numerical implementation method for the continuum model based on the level set representation of the dislocation array surfaces, which has the advantage of automatically handling the topological changes occurring during the evolution. Simulations are performed to examine the long-range nature of the interactions of dislocation arrays.

Randomized direct solvers for large discretized PDEs

Jianlin Xia

Purdue University, USA ([email protected] )

Abstract We propose some randomized structured direct solvers for large discretized PDEs, such as Helmholtz equations and systems arising from spectral methods. New randomization and low-rank techniques are used, together with flexible methods to exploit structures in large discretized matrices. Our randomized structured techniques provide both higher efficiency and better applicability than some existing structured methods. New efficient ways are proposed to conveniently perform various complex operations which are difficult otherwise. We also study the issues that are closely related, such as matrix-free direct solvers, frequency update, preconditioning, sparse eigenvalue solution, etc. The methods are especially useful for sparsifying the matrices resulting from spectral solutions and for solving 3D problems with varying parameters such as frequencies. We shown that they are relatively insensitive to the parameters and conditioning.

The study of the critical perturbation value for

singularly perturbed semilinear elliptic problems

Ziqing Xie

College of Mathematics and Computer Science, Hunan Normal University



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Abstract Inspired by our numerical observations, we prove rigorously the critical perturbation values which are thresholds to determine the existence of the nontrivial positive or negative solutions for some singularly perturbed semilinear elliptic problems. Some interesting numerical results are also shown.

Single-Grid Multilevel Method

Jinchao Xu

The Pennsylvania State University ([email protected])

Abstract In this talk, I will present a new approach to designing multigrid methods for discretized PDEs discretized on general unstructured grids, which we refer to as the "single-grid multilevel method" (SGML). The main idea is to use the information from the finest grid to select a simple and fixed coarsening that allows explicite control of the overall grid and operator complexities of the multilevel solver. The main issue to address is parallelization.

Dirichlet-Neumann Operator and Robin-type Domain

Decomposition

Xuejun Xu

LSEC, Institute of Computational Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing

Abstract In this talk, the tight relationship between Dirichlet-Neumann (D-N) operators and optimized Schwarz methods with Robin transmission conditions is disclosed. We shall describe the spectral distribution of the continuous and discrete D-N operators and give a rigorous spectral analysis of the Robin-type domain decomposition method. An optimal Robin-type domain decomposition shall be presented.



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Energy stable numerical schemes and simulations for two

phase complex fluids on the phase field method

Xiaofeng Yang Department of Mathematics, University of South Carolina

([email protected])

Abstract We present an energetic variational phase-field model for the two-phase Incompressible flow with one phase being the nematic liquid crystal. The model leads to a coupled nonlinear system satisfying an energy law. An efficient and easy-to-implement numerical scheme is presented for solving the coupled nonlinear system. We use this scheme to simulate two benchmark experiments: one is the formation of a bead-on-a-string phenomena, and the other is the dynamics of drop pinching-off. We investigate the detailed dynamical pinch-off behavior, as well as the formation of the consequent satellite droplets, by varying order parameters of liquid crystal bulk and interfacial anchoring energy constant. Qualitative agreements with experimental results are observed.

The small Deborah number limit of the Doi-Onsager

equation to the Ericksen-Leslie equation

Pingwen Zhang School of Mathematical Sciences, Peking University

([email protected])

Abstract We present a rigorous derivation of the Ericksen-Leslie equation starting from the Doi-Onsager equation. As in the fluid dynamic limit of the Boltzmann equation, we first make the Hilbert expansion for the solution of the Doi-Onsager equation. The existence of the Hilbert expansion is connected to an open question whether the energy of the Ericksen-Leslie equation is dissipated. We show that the energy is dissipated for the Ericksen-Leslie equation derived from the Doi-Onsager equation. The most difficult step is to prove a uniform bound for the remainder in the Hilbert expansion. This question is connected to the spectral stability of the linearized Doi-Onsager operator around a critical point. By introducing two important auxiliary operators, the detailed spectral information is obtained for the linearized operator around all critical points. However, these are not enough to justify the small Deborah number limit for the inhomogeneous Doi-Onsager equation, since the elastic stress in the velocity equation is also strongly singular. For this, we need to establish a precise lower bound for a bilinear form associated with the linearized operator. In the bilinear form, the interactions between the part inside the kernel and the part




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outside the kernel of the linearized operator are very complicated. We find a coordinate transform and introduce a five dimensional space called the Maier-Saupe space such that the interactions between two parts can been seen explicitly by a delicate argument of completing the square. However, the lower bound is very weak for the part inside the Maier-Saupe space. In order to apply them to the error estimates, we have to analyze the structure of the singular terms and introduce a suitable energy functional.

Superconvergence: Unclaimed Territories (Part 2)

Zhimin Zhang Wayne State University, USA

([email protected])

Abstract While the superconvergence phenomenon of the traditional Galerkin finite element method is well understood, we know very little, if any, about superconvergence for the spectral method and spectral collocation method. In this talk, we will touch this "unclaimed" territory by discussing superconvergence phenomenon of the polynomial spectral collocation method. In Part 2, the emphasis is on while interpolating the derivative of a targeting function, where should we evaluate the function value? in other words, where do the function value superconverges?

Symmetry Based Eigenvalue Computations

Aihui Zhou LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing,

Academy of Mathematics and Systems Science, Chinese Academy of Sciences ([email protected])

Abstract In this presentation, we will talk about a symmetry based numerical method for solving an eigenvalue problem, which divides an eigenvalue problem into group of smaller ones that can be solved independently. This method is straightforward and convenient for large-scale eigenvalue computation. Based on a finite element discretization of Kohn-Sham equation, we show particularly that this method can work well for both the Abelian and non-Abelian point group of molecular system. This presentation is based on a joint work with J. Fang and X. Gao.




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Abstract of Mini-symposium Talks

Uniform error estimates of numerical methods for

nonlinear Schrodinger equation with wave operator

Yongyong Cai Department of Mathematics, National University of Singapore

([email protected])

Abstract We propose an exponential wave integrator sine pseudospectral (EWI-SP) method for the nonlinear Schrodinger equation (NLS) with wave operator (NLSW), and carry out rigorous error analysis. NLSW is NLS perturbed by the wave operator with strength described by a

dimensionless parameter [0,1]ε ∈ .As 0ε +→ , NLSW converges to NLS and for the small

perturbation ,i.e. 0 1ε< < , the solution of NLSW differs from that of NLS with a function

oscillating in time with 2( )O ε -wavelength at 2( )O ε and 4( )O ε amplitudes for ill-prepared

and well-prepared initial data, respectively. This high oscillation in time brings significant difficulties in designing and analyzing numerical methods with error bounds uniformly in ε. In this work, we show that the proposed EWI-SP possesses the optimal uniform error bounds at

2( )O τ and ( )O τ in τ (time step) for well-prepared initial data and ill-prepared initial data,

respectively, and spectral accuracy in h (mesh size) for the both cases, in the L2 and semi- 1H norms. This result greatly improves the error bounds of the finite difference methods for NLSW in our recent work. Our approach involves a careful study of the error propagation, cut-off of the nonlinearity and the energy method. Numerical examples are provided to confirm our theoretical analysis.

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Superconvergence Properties of Variable-degree HDG

Methods for Convection-diffusion Equations on

Nonconforming Meshes.

Yanlai Chen University of Massachusetts Dartmouth

([email protected])

Abstract We present the error analysis of hybridizable discontinuous Galerkin (HDG) methods for




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convection-diffusion equations with variable-degree approximations on nonconforming meshes. In particular, for approximations of degree k on all elements and conforming meshes, we show that the order of convergence of the error in the diffusive flux is k+1 and that that of a projection of the error in the scalar unknown is 1 for k=0 and k+2 for k > 0. We also show that, for the variable-degree case, the projection of the error in the scalar variable is $h$-times the projection of the error in the vector variable. When general nonconforming meshes are used, our estimates do not rule out a degradation of 1/2 in the order of convergence in the diffusive flux and a loss of 1 in the order of convergence of the projection of the error in the scalar variable. They do guarantee the optimal convergence of order k+1 of the scalar variable. However, we show these losses of orders can be recovered if semimatching nonconforming meshes are used. These results hold for any (bounded) irregularity index of the nonconformity of the mesh. Finally, our analysis can be extended to hypercubes.

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Solving Surface PDEs via the Closest Point Method

Yujia Chen & Colin Macdonald Oxford Center for Collaborative Applied Mathematics (OCCAM), The University of Oxford

Abstract

This talk concerns the numerical solution of partial differential equations (PDEs) posed on general smooth surfaces by the Closest Point Method. The method extends the original surface PDE to a PDE defined in a narrow band surrounding the surface; then one can numerically solve the embedding PDE by finite difference schemes on Cartesian grids. I will give a description of the closest point method and some of its applications. I will also discuss some recent work on effectively solving the resulting sparse linear systems by a geometric multigrid method.

Numerical analysis for nonlinear Klein-Gordon equation

in the nonrelativistic limit regime

Xuanchun Dong Department of Mathematics, National University of Singapore

([email protected])

Abstract In this talk, we will consider the numerical solutions to nonlinear Klein-Gordon (KG) equation in a scaling involving a small parameterε . When 0 1ε< < , i.e. in the nonrelativistic limit regime, the solutions are highly oscillatory in time with a wave-length 2( )O ε . We will first show that the meshing strategy requirement (ε -scalability) for frequently used second-order finite-difference time integrator is τ = 3( )O ε (τ refers to time step). Later, we shall propose a class of exponential



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wave integrator for time advances, and rigorously prove that its ε -scalability is improved to τ = 2( )O ε . Last, we focus on the degenerate second order ODE arising in the semi-discretization of KG equation. For this highly oscillatory ODE, we will design two multi-scale numerical integrators based on two classes of wave decompositions, and rigorously prove that their error bounds are uniform in ε , i.e. their ε -scalability is τ = 2( )O ε for any 0 1ε< < . Numerical results are reported for better understanding the problem.

_____________________________________________________________________

Numerical computation of the electromagnetic

scattering from a two-dimensional open cavity

Kui Du School of Mathematical Sciences, Xiamen University

([email protected])

Abstract We consider the numerical computation of the electromagnetic scattering from a

two-dimensional open cavity embedded in the infinite ground plane. Transparent boundary conditions for both overfilled and non-overfilled cavities are discussed. For the non-overfilled case, we introduce several preconditioning techniques developed in recent years. Numerical results are reported to demonstrate the efficiency of the preconditioning techniques.

_____________________________________________________________________

A method for numerical analysis of a Lotka-Volterra

food web model

Jianlong Han Department of Mathematics, Southern Utah University

([email protected])

Abstract We study the existence, uniqueness and continuous dependence on initial data of the solution for a Lotka-Volterra cascade model with one basal species and hierarchal predation. A uniquely solvable, stable, semi-implicit finite-difference scheme is proposed for this system that converges to the true solution uniformly in a finite interval. This is based on work done with S.Armstrong. _____________________________________________________________________




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Adaptive PML finite element method for multiple

scattering problems

Xue Jiang LSEC, Institute of Computational Mathematics, Academy of Mathematics and systems science,

CAS ([email protected])

Abstract A uniaxial perfectly matched layer (UPML) method is proposed for solving three-dimensional Helmholtz equation with multiple scatterers. The exterior of each scatterer is truncated to a bounded domain and a PML absorbing layer is constructed to damp the outgoing waves from this scatterer. Based on conforming finite element approximation of the PML problem, adaptive UPML algorithms are developed with respect to reliable a posteriori error estimate. Numerical experiments show that the alternative UPML algorithm is economic and efficient on adaptively refined meshes.

FFTs on the Hexagonal and FCC lattices on GPUs

Huiyuan Li

Institute of Software, Chinese Academy of Sciences ([email protected] )

Abstract In this talk, we first give an introduction to the fast Fourier Transform (FFT) on lattices. By

representing the integer multi-index properly, a divide-and-conquer process is developed for the lattice FFTs. This process is finally reduced to a variety of the Cooley-Tukey algorithms which share a computational complexity of ( log )O N N .Then some CUDA algorithms on one single GPU are designed for FFTs on hexagonal and FCC lattices by utilizing the hierarchical parallelization mechanism. In contrast to the classic FFT, FFT on hexagonal and FCC lattices is essentially non-tensorial, and it can not be implemented by applying one-dimensional FFTs successively along each direction owing to its lack of linear separability. As a result, a great effort is made to devise and then optimize the algorithms for FFTs on hexagonal and FCC lattices on a GPU cluster. Finally, numerical experiments, which show a significant speedup of our algorithms, are presented.




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Development of finite point method for partial

differential equations I

Guixia Lv & Longjun Shen & Shunkai Sun National Key Laboratory of Science and Technology on Computational Physics, IAPCM

([email protected])

Abstract This talk focuses on a new meshless method —the finite point method(FPM). This methods is based on directional difference on scattered points. Different with many other meshless methods, FPM only needs 5 neighbouring scattered points in 2D to approximate directional differential with expected accuracy. We shall discuss the relationships between the multi-directional differentials of different orders in 2D. Based on these relationships, explicit formula are obtained for first order directional derivatives with second-order accuracy and second order directional derivatives with first-order accuracy. A sufficient and necessary condition for the solvability of the five-point formula is given. Methods for selecting permissible neighbouring point set which satisfy such a condition will be discussed. These methods are adapted to different problems.

Development of finite point method for partial

differential equations II

Guixia Lv & Longjun Shen & Shunkai Sun

National Key Laboratory of Science and Technology on Computational Physics, IAPCM

([email protected])

Abstract In this talk, we shall discuss how to use finite point method to solve kinds of PDEs on scattered data with multi-physics feature. Discrete schemes are designed for (non-linear) diffusion equations. Such schemes satisfy maximum principles of discrete solution to Laplace’s equations. Diffusion equations with multimedia are also in consideration; precisely, scattered points are placed on the multimedia interface, finite point scheme for the interface points are constructed. Numerical results show that these methods are second-order accurate for discrete solutions and first-order accurate for first-order derivatives. These methods can solve problems on irregular domain and on scattered points with large deformation. We shall also discuss how to use FPM to solve 1D and 2D compressible multifuids. A Lagrangian finite point method is presented. The proposed method is a meshfree numerical procedure based on the combination of interior points scheme and interface point tracking algorithm. The




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discretization of the unknown function and its derivatives are defined only by the position of the so called Lagrangian points. Numerical tests will demonstrate good performance of this method. We shall also discuss our progress on FPM methods for 3D problems if time permit.

Sampling with Localized and Weakly-localized Frames

Guohui Song Clarkson University

Abstract Frames can be seen as a generalization of basis with over-completeness. In many applications, frames provide robust and stable representation of functions due to this over-completeness. For example, in signal processing applications, the redundancy of frames is beneficial if signals are suspected of not capturing certain pieces of information. A fundamental problem in the frame representation is to find the inverse frame operator. In practice, however, it is very difficult or even impossible to find its closed form in an infinite-dimensional space. We shall study the numerical approximation of the inverse frame operator for localized and weakly-localized frames.

Devising superconvergent hybridizable DG methods for

Stokes and elasticity problem

Ke Shi University of Minnesota

([email protected])

Abstract We propose a projection-based a priori error analysis of finite element methods for second-order elliptic problems. The analysis is unifying because it applies to a large class of methods including the hybridized version of most well-known mixed methods as well as several hybridizable discontinuous Galerkin (HDG) methods. The novelty of the approach is that it reduces the whole error analysis to the element-by-element construction of an auxiliary projection satisfying certain orthogonality and approximation properties, and to the verification of very simple inclusion properties of the local spaces defining the methods. We extend this approach to Stokes equations and linear elasticity.



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Quantized vortex dynamics in complex Ginzburg-landau

equation in bounded domain

Qinglin Tang Department of Mathematics and Center for Computational Science and Engineering,

National University of Singapore ([email protected])

Abstract In this talk, I will present efficient and accurate numerical methods for studying the quantized vortex dynamics and their interaction of the two-dimensional (2D) complex Ginzburg-Landau equation (CGLE) with a dimensionless parameter ε > 0 in bounded domains under either Dirichlet or homogeneous Neumann boundary condition. I will begin with a review of various reduced dynamical laws for time evolution of quantized vortex centers and show how to solve these nonlinear ordinary differential equations numerically. Then, I will present some results on the quantized vortex interaction under various different initial setup, and identify the cases where the reduced dynamical laws agree qualitatively and/or quantitatively as well as fail to agree with those from CGLE. Some other interesting phenomena will also be presented.

An adaptive level set method based on two-level uniform

meshes and its application to dislocation dynamics

Hanquan Wang School of Statistics and Mathematics, Yunnan University of Finance and Economics

([email protected])

Abstract Dislocations are line defects and the primary carriers of plastic deformation in crystalline materials. The level set method based on uniform mesh has been employed in dislocation dynamics simulations to account for topological changes and various types of motion and interactions associated with the evolution of dislocation microstructures. The three dimensional dislocation dynamics is a co-dimension two problem, and the available techniques for improving the efficiency and accuracy of the level set method such as the narrow band or local level set methods and the tree-based adaptive level set methods do not apply directly. In this paper, we present an adaptive level set method for dislocation dynamics as well as other problems of motion of high co-dimensional objects. This method uses only two (or a few fixed) levels of meshes, and the meshes are uniform over the computational domain and in the refinement. The coarse-to-fine ratios in the mesh refinement can be adjusted to achieve optimal efficiency. In this adaptive




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method, the computation is localized 3 mostly near the moving objects, thus the computational cost is significantly reduced compared with the uniform mesh over the whole domain with the same resolution. In this method, the level set equations can be solved on these uniform meshes of different levels directly using standard high order numerical methods without involving the tree data structures. This two-level adaptive method also provides a basis for using locally varying time stepping to further reduce the computational cost. This is a joint work with Prof. Yang Xiang from Hongkong University of Science and Technology.

Asymptotic Analysis of Phase Field Formulations of

Bending Elasticity Models

Xiaoqiang Wang ([email protected])

Abstract In this talk, we will analysis the phase field formulation of the Bending Elasticity Models used in describing cell membranes. Our asymptotic analysis proves the consistency of the phase filed formulations to the sharp interface models. And based on the asymptotic analysis, we will also give the error estimates for several phase field formulae broadly used in modeling cell membranes.Numerical experiments will be used to illustrate our results.

A multi-scale moving boundary mathematical model for

cancer invasion

Yangfang Wang Ocean University of China

Abstract In this paper we introduce a new type of multiscale model for cancer invasion. This multi-scale model is developed by accounting the macroscopic dynamics of the distributions of cancer cells and of the surrounding extracellular matrix (ECM), which is taking place on a macroscopic domain that is permanently expanded and moved by a matrix metalloproteinase micro-dynamics developed on the microscopic-scale spatial-neighbouring bundle of its boundary. While the macroscopic process is describing the dynamics of the spatio-temporal distribution of the cancer cells and ECM on an evolving macro-domain Ω(t0), the microscopic boundary processes are expressing the molecular dynamics for bound MMP molecular population that are produced from within the cancer cells distribution on the macroscopic domain, which, around each point of the



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boundary ∂Ω(t0), are locally diffused into the surrounding tissue outside the cancer cluster. The macroscopic process regarding the dynamics of the cancer cells and the ECM enables the exploration and extensive mathematical characterisation of the source of bound MMP that occur in each of the boundary micro-processes and facilitates their micro-dynamics. Further, within the induced bundle of boundary micro-processes, a naturally inherited micro-stochasticity is detected and mathematically formalised. By combining this naturally induced stochasticity with the dynamics of the associated micro-processes, we obtain a mathematical characterisation of the evolution of the boundary of the macro-domain ∂Ω(t0) towards a new boundary corresponding to the expanded macro-domain Ω(t0) + ∆t, where the cancer process continues with its next stage of invasion.

Pre-asymptotic error Analysis of CIP-FEM and FEM for

Helmholtz Equation with high Wave Number

Haijun Wu Department of Mathematics, Nanjing University

([email protected])

Abstract The continuous interior penalty finite element method (CIP-FEM) and the FEM for the Helmholtz equation in two and three dimensions are considered. By using a modified duality argument, pre-asymptotic error estimates are derived for both methods under the condition of

1/( 1)0 ( ) pkh pC

p k+< , where k is the wave number, h is the mesh size, and 0C is a constant

independent of , ,k h p , and the penalty parameters. It is shown that the pollution errors of both

methods in 1H -norm are 2 1 2( )p pO k h+ if p=O(1) and are 22( ( ) )pk kO p

p hσ if the exact solution

2 ( )u H∈ Ω which coincide with existent dispersion analyses for the FEM on Cartesian grids. Here σ is a constant independent of k, h, p, and the penalty parameters. Moreover, it is proved that the CIP-FEM is stable for any k, h, p>0 and penalty parameters with positive imaginary parts. Besides the advantage of the absolute stability of the CIP-FEM compared to the FEM, the penalty parameters may be tuned to reduce the pollution effects.

Negative-Order Norm Estimates for Nonlinear Hyperbolic

Conservation Laws

Yan Xu University of Science and Technology of China



45

([email protected])

Abstract In this paper, we establish negative-order norm estimates for the accuracy of discontinuous Galerkin (DG) approximations to scalar nonlinear hyperbolic equations with smooth solutions. For these special solutions, we are able to extract this “hidden accuracy” through the use of a convolution kernel that is composed of a linear combination of B-splines. Previous investigations into extracting the superconvergence of DG methods using a convolution kernel have focused on linear hyperbolic equations. However, we now demonstrate that it is possible to extend the smoothness-increasing accuracy-conserving (SIAC) filter for scalar nonlinear hyperbolic equations. Furthermore, we provide theoretical error estimates that show improvement to (2k+m)-th order in the negative-order norm, where m depends upon the chosen flux.

Superconvergence of discontinuous Galerkin method for

linear hyperbolic equations in one space dimension

Yang Yang Brown University

([email protected])

Abstract We study the superconvergence of the error between the discontinuous Galerkin (DG) finite element solution and the exact solution for linear conservation laws when upwind fluxes are used. We prove that if we apply piecewise k-th degree polynomials, the error between the DG solution and the exact solution is (k + 2)-th order superconvergent at the downwind-biased Radau points with suitable initial discretization. Moreover, we also prove the DG solution is (k + 2)-th order superconvergent both for the cell averages and for the error to a particular projection of the exact solution. The proof is valid for arbitrary regular meshes and for kp polynomials with arbitrary k ≥ 1, and for both periodic boundary conditions and for initial-boundary value problems. We provide numerical experiments of polynomials of degree m = 1 and 2 to demonstrate that the convergent rate is optimal.

Numerical Study of the Instability of Incompressible

Channel Flows

Haijun Yu

Institute of Computational Mathematics, CAS ([email protected])




46

Abstract Highly accurate spectral methods are used to calculate the nontrivial solutions of incompressible planar channel flows. Direct numerical simulation with stochastic forces adopted to evaluate the stabilities of those trivial and nontrivial solutions. A sketch of "phase diagram" together with explanations will be presented in this talk.

Self-consistent Mean Field Model of Hydrogel and Its

Numerical Simulation

Hui Zhang Department of Mathematics, Beijing Normal University, China

([email protected])

Abstract A model to describe the micro-structure of Macromolecular Microsphere Composite (MMC) Hydrogel is proposed in the framework of self-consistent mean field theory (SCFT). The equations systems derived by SCFT approximation are solved by a relax algorithm efficiently and effectively. From SCFT simulations of the model, we find that two parameters play an important role on the whole procedure of the micro-structure of MMC hydrogel, the interactions between two species: polymer chains and MMS spheres, and the volume fractions of MMS spheres. The role of various parameters on the structure is also discussed in this paper. It is found that simulation results are partly consistent with the observation from the experiments. Moreover, we also show some new microstructures.

Consistent Atomistic/Continuum Coupling

Lei Zhang Shanghai Jiontong Univeristy//The University of Oxford

([email protected])

Abstract Atomistic/continuum coupling methods (A/C methods) are a class of coarse-graining techniques for the efficient simulation of atomistic systems with localized defects interacting with long-range




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elastic field. In this talk, I will show that for general simple crystal lattices, patch test consistency is a necessary and sufficient condition for the first-order consistency of an A/C method. Moreover, we will construct consistent methods by geometric reconstruction, which is the first such construction in 2D and 3D. In the end, I will remark on the stability and a priori error estimate of the method.

Noise Drives Sharpening of Expression Boundaries in the

Zebrafish Hindbrain

Lei Zhang Department of Mathematics, University of California at Irvine

([email protected])

Abstract Morphogens provide positional information for spatial patterns of gene expression during development. However, stochastic effects such as local fluctuations in morphogen concentration and noise in signal transduction make it difficult for cells to respond to their positions accurately enough to generate sharp boundaries between gene expression domains. In this talk, I will present a multiscale stochastic model to investigate a novel noise attenuation mechanism during the development in the zebrafish hindbrain. Computational analyses of spatial stochastic models show, surprisingly, that a combination of noise in RA concentration and noise in hoxb1a/krox20 expression promotes sharpening of boundaries between adjacent segments. In particular, fluctuations in RA initially induce a rough boundary that requires noise in hoxb1a/krox20 expression to sharpen. This finding suggests a novel noise attenuation mechanism that relies on intracellular noise to induce switching and coordinate cellular decisions during developmental patterning.

Multiscale eddy current problems

Weiying Zheng Academy of Mathematics and Systems Science, Chinese Academy of Sciences

([email protected])

Abstract In a large power transformer, the iron core and the magnetic shields of the oil tank are usually made of grain-oriented silicon steel laminations. The steel laminations and their coating films are very thin so that the ratio of the largest scale of the device to the smallest scale can amount to 106.




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The nonlinear eddy current model of Maxwell’s equations is used to compute the iron loss of conductors. In this lecture, I shall talk about the mathematical modeling of the eddy current problem to reduce the small scales. First we propose a simplified model to compute 3D eddy currents in the laminations by omitting coating films. Then we propose a macro-scale eddy current model by using homogenization method.

Smoothness matching for radial basis functions

Shengxin Zhu & Rosemary A. Renaut & Andy J. Wathen (Oxford Center for Collaborative Applied Mathematics(OCCAM), The University of Oxford)

Abstract Radial basis functions have attracted a lot of attention in recent years as an elegant approximation scheme for high-dimensional scattered data approximation, one foundation of mesh-free method and an emerging method for surface computing. We shall discuss some fundamental issues related to effective computing with radial basis functions: smoothing effect of kernel matrices related to radial basis function. And we shall show how this issue impacts on smoothness matching between radial basis function and target functions.


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Abstract of Contributed Talks/Posters

Parallel spectral element method using direction

splitting for the incompressible Navier-Stokes

equations

Lizhen Chen Beijing Computational Science Research Center

([email protected])

Abstract: An efficient parallel spectral element method for the incompressible Navier-Stokes equations is developed.

Using the direct splitting method, the standard Poisson equation for the pressure correction is replaced by

a series of one-dimensional second-order boundary boundary value problems that can be efficiently

parallelized. In the space discretization, we use the Legendre-galerkin spectral element method.

Numerical tests show that this method keeps the spectral accuracy and our algorithm achieves very high

parallel efficiency, so this method can be widely used in computational fluid dynamics and computational

material science

Loosely coupled parallel computation of an SVD and

applications in optimization and machine learning

Sheng Fang The University of Oxford ([email protected])

Abstract Multicore processors and graphics cards are fast becoming standard features of personal

computers. This additional computational power presents huge opportunities for scientific computing, but harnessing it is nontrivial, for it requires the development of new numerical linear algebra algorithms based on loosely coupled parallelism (low synchronicity, low communication overhead). In this talk we present such an algorithm for the computation of the leading part singular value decomposition of very large matrices, and we discuss the uses of this approach in continuous optimization models

_____________________________________________________________________




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A Coupling of Local Discontinuous Galerkin and Natural

Boundary Element Method for Exterior Problems

Hongying Huang School of Mathematics, Physics and Information Science,

Zhejiang Ocean University ([email protected])

Abstract In this paper, we apply the coupling of local discontinuous Galerkin (LDG) and natural boundary element method(NBEM) to solve a two-dimensional exterior problem. As a consequence, the main features of LDG and NBEM are maintained and hence the coupled approach benefits from the advantages of both methods. Referring to Gatica et al. (Math. Comput. 79(271):1369–1394, 2010), we employ LDG subspaces whose functions are continuous on the coupling boundary. In this way, the primitive variables become the only boundary unknown, and hence the total number of unknown functions is reduced. We prove the stability of the new discrete scheme and derive an a priori error estimate in the energy norm. Some numerical examples conforming the theoretical results are provided.

A stable algorithm for non-homogeneous waveguide

equation based on DtN maps

Jinyang Huang College of Science, Beijing University of Chemical Technology

([email protected])

Abstract A new stable computational method for non-homogeneous waveguide equation with a piecewise uniform structure along the main propagation direction is constructed, based on the modified Dirichlet-to-Neumann (DtN) map of each uniform segment. For segments with the same structure, only a DtN map needs to be calculated on such a segment, and then the solution of the equation can be derived recursively. Numerical examples demonstrate that it is a stable and efficient algorithm for the waveguide equations. This method can greatly reduces the requirement of internal memory and the amount of computation compared with the traditional algorithms.


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Full Eulerian modeling and effective numerical studies

for the dynamic fluid-structure interaction problem

Pengtao Sun University of Nevada Las Vegas

([email protected])

Abstract In this paper we present a full Eulerian model for a fluid-structure interaction (FSI) problem in terms of phase field approach, and design its effective numerical discretizations and iterative schemes based on the second-order backward difference formula (BDF), a combined finite element-upwind finite volume method, Galerkin/least-square (GLS) stabilization scheme as well as streamline-upwind/Petrov-Galerkin (SUPG) approach. The presented FSI model can preserve the shape and the size of the structure frame during its rotation while interacting with the fluid that is flowing around. Numerical experiments done for some different structure frames strongly show that our model and numerical methods are effective to simulate the dynamic fluid-structure interaction phenomena. This work is joint with Lixiang Zhang, Jinchao Xu and Chun Liu.

Numerical stability of split-step forward Euler method

for stochastic delay differential equations

Qi Wang Guangdong University of Technology

([email protected])

Abstract For linear stochastic delay differential equations with multiplicative noise, mean square stability and T-stability of drifting split-step Euler (DRSSE) method which belongs to the split-step forward Euler method are both studied. Under certain condition for coefficients, it is proven that the numerical solution is mean square stable when the step-size satisfies certain restrictions. Moreover, by discussing the difference equation, which is the outcome of applying the numerical method with a specified driving process to the given equation, the sufficient conditions of T-stability are given.



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The dynamics of quantized vortices in Ginzburg-Landau

Equation based on particle interaction

Zhiguo Xu Beijing Computational Science Research Center

([email protected])

Abstract

Ginzburg-Landau equation (GLE) is well known for modeling superconductivity. In this paper, a review of the reduced dynamic law (ODEs) are governing the motion of vortices in the GLE is provided. Based on the reduced dynamic law, we study the dynamics of quantized vortices in GLE. We find some conserved quantities of the reduced dynamic law. According to these conserved quantities, some general results are given. Particularly, we show the dynamics of three vortices completely including the long time behavior and collision of vortices. Finally, we study the total collision of vortices.

A Fast Accurate Boundary Integral Method for Potentials

on Closely Packed Cells

Wenjun Ying Department of Mathematics and Institute of Natural Sciences,

Shanghai Jiao Tong University

Abstract Boundary integral methods are naturally suited for the computation of harmonic functions on a region having inclusions or cells with different material properties. However, accuracy deteriorates when the cell boundaries are close to each other. We present a boundary integral method which is specially designed to maintain second order accuracy even if boundaries are arbitrarily close. The method uses a regularization of the integral kernel which admits analytically determined corrections to maintain accuracy. For boundaries with many components we use the fast multipole method for efficient summation. We compute electric potentials on a domain with cells whose conductivity differs from that of the surrounding medium. We first solve an integral equation for a source term on the cell interfaces and then find values of the potential near the interfaces via integrals. Finally we use a Poisson solver to extend the potential to a regular grid covering the entire region. A number of examples are presented. We demonstrate that increased refinement is not needed to maintain accuracy as interfaces become very close. This is a joint research work with J. Thomas Beale, Duke University.



53

A convergent adaptive finite element algorithm for

nonlocal diffusion models

Xuying Zhao AMSS, Chinese Academy of Sciences

([email protected])

Abstract Nonlocal models such as nonlocal diffusion equations and nonlocal peridynamic models have attracted much attention recently. In this paper, we propose an adaptive finite element algorithm for the numerical solution of a class of nonlocal models which correspond to nonlocal diffusion equations with certain non-integrable kernel functions. The convergence of the adaptive finite element algorithm is rigorously derived with the help of several basic ingredients, such as the upper bound of the estimator, the estimator reduction and the orthogonality property. We also consider how the results are affected by the horizon parameter which characterizes the range of nonlocality. Numerical experiments are performed to verify our theoretical findings. It is a joint work with Prof. Qiang Du and Dr. Li Tian.

Cross entropy method for multiple modes in rare-event

simulation

Xiang Zhou Division of applied Math, Brown University

([email protected])

Abstract Importance sampling is a useful tool to estimate the extremely low probability of rare events. Cross entropy is a universal method to find a good importance sampling measure by minimizing the relative distance to the best ideal change of measure. However, in the case of existence of multiple local minimizers for all non-convex problems, an optimal mixing of multiple probability measures is necessary and we propose a new simple-to-implement and efficient algorithm to solve this problem. This is joint work with Hui Wang.




54

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Program and Abstract Book - Xiamen Universitymath.xmu.edu.cn/conference/Brochure4.pdf · Xue-Cheng Tai: A fast algorithm ... A method for numerical analysis of a Lotka -Volterra food