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  • Finite Element Multigrid Framework for Mimetic Finite Difference Discretizations

    Xiaozhe Hu

    Tufts University

    Polytopal Element Methods in Mathematics and Engineering, October 26 - 28, 2015

    Joint work with: F.J. Gaspar, C. Rodrigo (Universidad de Zaragoza),

    and L. Zikatanov (Penn State)

    X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 1 / 25

  • Outline

    1 Introduction

    2 Relation Between Finite Element and Mimetic Finite Difference

    3 Geometric Multigrid Methods

    4 Conclusions and Future Work

    X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 2 / 25

  • Introduction

    Outline

    1 Introduction

    2 Relation Between Finite Element and Mimetic Finite Difference

    3 Geometric Multigrid Methods

    4 Conclusions and Future Work

    X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 3 / 25

  • Introduction

    Model Problems:

    Model Equations

    curl rotu + κu = f, in Ω

    −grad divu + κu = f, in Ω

    • Applications: Darcy’s flow, Maxwell’s equation, etc. • Involve special physical and mathematical properties: mass conservation,

    Gauss’s Law, exact sequence property of the differential operators, etc. • Complicated geometry: unstructured triangulation, polytopal mesh, etc.

    Structure-preserving discretizations on polytopal meshes are preferred!!

    • Mimetic finite difference method (Lipnikov, Manzini, & Shashkov 2014; Beirão Da Veiga, Lipnikov, & Manzini 2014;...)

    • Generalized finite difference method (Bossavit 2001; 2005; Gillette & Bajaj 2011; ...) • Mixed finite element method (Brezzi & Fotin 1991; ...) • Finite element exterior calculus (Arnold, Falk, & Winther 2006; 2010; ...) • Discontinuous Galerkin method (Arnold, Brezzi, Cockburn, & Marini 2002; ...) • Virtual element method (Beirão Da Veiga, Brezzi, Cangiani, Manzini, Marini & Russo 2013; ...) • Weak Galerkin method (Wang & Ye 2013; ...) • Hybrid High-Order method (Di Pietro, Ern, & Lemaire 2014; ...) • ...

    X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 4 / 25

  • Introduction

    Model Problems:

    Model Equations

    curl rotu + κu = f, in Ω

    −grad divu + κu = f, in Ω

    • Applications: Darcy’s flow, Maxwell’s equation, etc. • Involve special physical and mathematical properties: mass conservation,

    Gauss’s Law, exact sequence property of the differential operators, etc. • Complicated geometry: unstructured triangulation, polytopal mesh, etc.

    Structure-preserving discretizations on polytopal meshes are preferred!!

    • Mimetic finite difference method (Lipnikov, Manzini, & Shashkov 2014; Beirão Da Veiga, Lipnikov, & Manzini 2014;...)

    • Generalized finite difference method (Bossavit 2001; 2005; Gillette & Bajaj 2011; ...) • Mixed finite element method (Brezzi & Fotin 1991; ...) • Finite element exterior calculus (Arnold, Falk, & Winther 2006; 2010; ...) • Discontinuous Galerkin method (Arnold, Brezzi, Cockburn, & Marini 2002; ...) • Virtual element method (Beirão Da Veiga, Brezzi, Cangiani, Manzini, Marini & Russo 2013; ...) • Weak Galerkin method (Wang & Ye 2013; ...) • Hybrid High-Order method (Di Pietro, Ern, & Lemaire 2014; ...) • ...

    X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 4 / 25

  • Introduction

    Model Problems:

    Model Equations

    curl rotu + κu = f, in Ω

    −grad divu + κu = f, in Ω

    • Applications: Darcy’s flow, Maxwell’s equation, etc. • Involve special physical and mathematical properties: mass conservation,

    Gauss’s Law, exact sequence property of the differential operators, etc. • Complicated geometry: unstructured triangulation, polytopal mesh, etc.

    Structure-preserving discretizations on polytopal meshes are preferred!!

    • Mimetic finite difference method (Lipnikov, Manzini, & Shashkov 2014; Beirão Da Veiga, Lipnikov, & Manzini 2014;...)

    • Generalized finite difference method (Bossavit 2001; 2005; Gillette & Bajaj 2011; ...) • Mixed finite element method (Brezzi & Fotin 1991; ...) • Finite element exterior calculus (Arnold, Falk, & Winther 2006; 2010; ...) • Discontinuous Galerkin method (Arnold, Brezzi, Cockburn, & Marini 2002; ...) • Virtual element method (Beirão Da Veiga, Brezzi, Cangiani, Manzini, Marini & Russo 2013; ...) • Weak Galerkin method (Wang & Ye 2013; ...) • Hybrid High-Order method (Di Pietro, Ern, & Lemaire 2014; ...) • ...

    X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 4 / 25

  • Introduction

    Model Problems:

    Model Equations

    curl rotu + κu = f, in Ω

    −grad divu + κu = f, in Ω

    • Applications: Darcy’s flow, Maxwell’s equation, etc. • Involve special physical and mathematical properties: mass conservation,

    Gauss’s Law, exact sequence property of the differential operators, etc. • Complicated geometry: unstructured triangulation, polytopal mesh, etc.

    Structure-preserving discretizations on polytopal meshes are preferred!!

    • Mimetic finite difference method (Lipnikov, Manzini, & Shashkov 2014; Beirão Da Veiga, Lipnikov, & Manzini 2014;...)

    • Generalized finite difference method (Bossavit 2001; 2005; Gillette & Bajaj 2011; ...) • Mixed finite element method (Brezzi & Fotin 1991; ...) • Finite element exterior calculus (Arnold, Falk, & Winther 2006; 2010; ...) • Discontinuous Galerkin method (Arnold, Brezzi, Cockburn, & Marini 2002; ...) • Virtual element method (Beirão Da Veiga, Brezzi, Cangiani, Manzini, Marini & Russo 2013; ...) • Weak Galerkin method (Wang & Ye 2013; ...) • Hybrid High-Order method (Di Pietro, Ern, & Lemaire 2014; ...) • ...

    X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 4 / 25

  • Introduction

    Motivation

    A question:

    How to solve Ax = b efficiently

    This talk:

    • focus on mimetic FDM (Vector Analysis Grid Operators Method, Vabishchevich, 2005) • show relation between mimetic FDM and FEM • design geometric multigrid methods for mimetic FDM

    Relation between MFD and MFEM for diffusion (Berndt, Lipnikov, Moulton, & Shashkov 2001;

    Berndt, Lipnikov, Shashkov, Wheeler & Yotov 2005; Droniou, Eymard, Gallouët, & Herbin 2010)

    X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 5 / 25

  • Introduction

    Motivation

    A question:

    How to solve Ax = b efficiently

    This talk:

    • focus on mimetic FDM (Vector Analysis Grid Operators Method, Vabishchevich, 2005) • show relation between mimetic FDM and FEM • design geometric multigrid methods for mimetic FDM

    Relation between MFD and MFEM for diffusion (Berndt, Lipnikov, Moulton, & Shashkov 2001;

    Berndt, Lipnikov, Shashkov, Wheeler & Yotov 2005; Droniou, Eymard, Gallouët, & Herbin 2010)

    X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 5 / 25

  • Introduction

    Motivation

    A question:

    How to solve Ax = b efficiently

    This talk:

    • focus on mimetic FDM (Vector Analysis Grid Operators Method, Vabishchevich, 2005)

    • show relation between mimetic FDM and FEM • design geometric multigrid methods for mimetic FDM

    Relation between MFD and MFEM for diffusion (Berndt, Lipnikov, Moulton, & Shashkov 2001;

    Berndt, Lipnikov, Shashkov, Wheeler & Yotov 2005; Droniou, Eymard, Gallouët, & Herbin 2010)

    X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 5 / 25

  • Introduction

    Motivation

    A question:

    How to solve Ax = b efficiently

    This talk:

    • focus on mimetic FDM (Vector Analysis Grid Operators Method, Vabishchevich, 2005) • show relation between mimetic FDM and FEM

    • design geometric multigrid methods for mimetic FDM

    Relation between MFD and MFEM for diffusion (Berndt, Lipnikov, Moulton, & Shashkov 2001;

    Berndt, Lipnikov, Shashkov, Wheeler & Yotov 2005; Droniou, Eymard, Gallouët, & Herbin 2010)

    X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 5 / 25

  • Introduction

    Motivation

    A question:

    How to solve Ax = b efficiently

    This talk:

    • focus on mimetic FDM (Vector Analysis Grid Operators Method, Vabishchevich, 2005) • show relation between mimetic FDM and FEM • design geometric multigrid methods for mimetic FDM

    Relation between MFD and MFEM for diffusion (Berndt, Lipnikov, Moulton, & Shashkov 2001;

    Berndt, Lipnikov, Shashkov, Wheeler & Yotov 2005; Droniou, Eymard, Gallouët, & Herbin 2010)

    X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 5 / 25

  • Introduction

    Motivation

    A question:

    How to solve Ax = b efficiently

    This talk:

    • focus on mimetic FDM (Vector Analysis Grid Operators Method, Vabishchevich, 2005) • show relation between mimetic FDM and FEM • design geometric multigrid methods for mimetic FDM

    Relation between MFD and MFEM for diffusion (Berndt, Lipnikov, Moulton, & Shashkov 2001;

    Berndt, Lipnikov, Shashkov, Wheeler & Yotov 2005; Droniou, Eymard, Gallouët, & Herbin 2010)

    X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 5 / 25

  • Introduction