Download pdf - Finite Element Multigrid Framework for Mimetic …...Finite Element Multigrid Framework for Mimetic Finite Di erence Discretizations Xiaozhe Hu Tufts University Polytopal Element Methods

Finite Element Multigrid Framework forMimetic 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


Introduction

Outline

1 Introduction





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; Beirao 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 (Beirao Da Veiga, Brezzi, Cangiani, Manzini, Marini & Russo 2013; ...)

• Weak Galerkin method (Wang & Ye 2013; ...)

• Hybrid High-Order method (Di Pietro, Ern, & Lemaire 2014; ...)

• ...


Introduction

Model Problems:

Model Equations















• ...


Introduction

Model Problems:

Model Equations















• ...


Introduction

Model Problems:

Model Equations















• ...


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, Gallouet, & Herbin 2010)


Introduction

Motivation

A question:


This talk:







Introduction

Motivation

A question:


This talk:







Introduction

Motivation

A question:


This talk:







Introduction

Motivation

A question:


This talk:







Introduction

Motivation

A question:


This talk:







Introduction

Mimetic FDM: Delaunay and Voronoi Grids

Computational domain:

Ω = Ω ∪ ∂Ω

Acute Delaunay grid

xDi , i = 1, . . . ,ND

Dual mesh: Voronoi grid

Voronoi points: centers of the circumscribedcircles on each triangle

xVk , i = 1, . . . ,NV

For each xDi

Voronoi polygon:

Vi = x ∈ Ω | |x− xDi | < |x− xDj |, j = 1, . . . ,ND , j 6= i,

and we denote: ∂Vij = ∂Vi ∩ ∂Vj


Introduction



Ω = Ω ∪ ∂Ω

Acute Delaunay grid

xDi , i = 1, . . . ,ND



xVk , i = 1, . . . ,NV

For each xDi

Voronoi polygon:




Introduction



Ω = Ω ∪ ∂Ω

D

D

Acute Delaunay grid

xDi , i = 1, . . . ,ND



xVk , i = 1, . . . ,NV

For each xDi

Voronoi polygon:




Introduction

Mimetic FDM: Grid Functions

• Scalar Grid Functions:

• Delaunay grid: u(x) are defined by u(xDi ) = uDi at the nodesxDi . HD denotes the set of u(x).

• Voronoi grid: u(x) are defined by u(xVk ) = uVk at the nodesxVk . HV denotes the set of u(x).

• Vector Grid Functions:

• Delaunay grid: u(x) are defined by u(x) · eDij = uDij at the

middle point of the edges xDij = 12 (xDi + xDj ). HD denotes the

set of u(x)

• Voronoi grid: u(x) are defined by u(x) · eVkm = uVkm at theintersect points. HV denotes the set of u(x)

eDij is directed from the node with smaller index to the node with larger index


Introduction








set of u(x)




Introduction








set of u(x)




Introduction

Mimetic FDM: Discrete Operators

Discrete Gradient Operators: gradh : HD → HD

(gradh u)Dij :=η(i , j)uDj − uD

i

lDij, with η(i , j) =

1, if j > i−1, if j < i

Discrete Rotor Operator: roth : HD → HV

(roth u)Vk =η(i , j) uD

ij lDij + η(j , l) uD

jl lDjl + η(l , i) uD

li lDli

meas(Dk)

Discrete Curl Operator: curlh : HV → HD

(curlh u)Dij = η(k,m)uVk − uV

m

lVkm

Discrete Divergence Operator: divh : HD → HD

(divh u)Di =1

meas(Vi )

∑j∈WV (i)

uDij (eDij · nV

ij ) meas(∂Vij)

D

D D

V

D

D

D

D

D

V V

V

D

D


Introduction




i


1, if j > i−1, if j < i





li lDli

meas(Dk)



m

lVkm


(divh u)Di =1

meas(Vi )

∑j∈WV (i)

uDij (eDij · nV

ij ) meas(∂Vij)

D

D D

V

D

D

D

D

D

V V

V

D

D


Introduction




i


1, if j > i−1, if j < i





li lDli

meas(Dk)



m

lVkm


(divh u)Di =1

meas(Vi )

∑j∈WV (i)

uDij (eDij · nV

ij ) meas(∂Vij)

D

D D

V

D

D

D

D

D

V V

V

D

D


Introduction




i


1, if j > i−1, if j < i





li lDli

meas(Dk)



m

lVkm


(divh u)Di =1

meas(Vi )

∑j∈WV (i)

uDij (eDij · nV

ij ) meas(∂Vij)

D

D D

V

D

D

D

D

D

V V

V

D

D


Introduction




i


1, if j > i−1, if j < i





li lDli

meas(Dk)



m

lVkm


(divh u)Di =1

meas(Vi )

∑j∈WV (i)

uDij (eDij · nV

ij ) meas(∂Vij)

D

D D

V

D

D

D

D

D

V V

V

D

D


Introduction

Mimetic FDM: Stencils

1/ 2/3

2/32/3

2/3

2/32/3

1/

4√3/3 4√3/3

4√3/3

√3/

√3/

gradh divh roth curlh

Mimetic FDM

curlh rothuh + κuh = fh, in Ω

−gradh divhuh + κuh = fh, in Ω

8/

4/

4/ 4/

4/

4/3 2/3 2/3

2/3 2/3

2/3 2/3

2/3

2/3 2/32/3

curlh roth − gradh divh


Introduction


1/ 2/3

2/32/3

2/3

2/32/3

1/

4√3/3 4√3/3

4√3/3

√3/

√3/


Mimetic FDM



8/

4/

4/ 4/

4/

4/3 2/3 2/3

2/3 2/3

2/3 2/3

2/3

2/3 2/32/3



Introduction


1/ 2/3

2/32/3

2/3

2/32/3

1/

4√3/3 4√3/3

4√3/3

√3/

√3/


Mimetic FDM



8/

4/

4/ 4/

4/

4/3 2/3 2/3

2/3 2/3

2/3 2/3

2/3

2/3 2/32/3



Relation Between Finite Element and Mimetic Finite Difference

Outline

1 Introduction





Relation Between Finite Element and Mimetic Finite Difference curlh roth

Mimetic FDM and Nedelec FEM

Nedelec FEM

Find uh ∈ VNh , such that,

(rot uh, rot vh) + κ(uh, vh) = (f, vh), ∀ vh ∈ VNh

where VNh consists lowest order Nedelec finite elements.

• Equilateral triangle:

Mimetic FDM Nedelec FEM

8/ℎ2

4/ℎ2

4/ℎ2 −4/ℎ2

−4/ℎ2

8√3/3ℎ2

4√3/3ℎ2

4√3/3ℎ2 −4√3/3ℎ2

−4√3/3ℎ2



Mimetic FDM and Nedelec FEM

Nedelec FEM

Find uh ∈ VNh , such that,

(rot uh, rot vh) + κ(uh, vh) = (f, vh), ∀ vh ∈ VNh

where VNh consists lowest order Nedelec finite elements.

• Equilateral triangle:


8/ℎ2

4/ℎ2

4/ℎ2 −4/ℎ2

−4/ℎ2

8√3/3ℎ2

4√3/3ℎ2

4√3/3ℎ2 −4√3/3ℎ2

−4√3/3ℎ2



Mimetic FDM and Modified Nedelec FEM• General triangle:

𝑙𝑗𝑙𝐷

𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)

−𝑙𝑖𝑙𝐷


𝑙𝑗𝑙𝐷


−𝑙𝑖𝑙𝐷


2𝑙𝑖𝑗𝐷


𝛼 𝛽

𝑥𝑖𝐷

𝑥𝑙𝐷

𝑥𝑗𝐷

1

𝑚𝑒𝑎𝑠(𝐷𝑘) −

1

𝑚𝑒𝑎𝑠(𝐷𝑘)

1


1


2


𝛼 𝛽


Consider a function u(x) ∈ VNh ,

u(x) =∑(i,j)

DOFNij (u)ϕij =

∑(i,j)

(∫ xDj

xDi

u · eDij

)ϕij

=∑(i,j)

(u · eDij )(xDij ) lDij︸︷︷︸

midpoint rule

ϕij =∑(i,j)

DOFMFDij (u)lDij ϕij

• Modified basis functions: ϕmodij = lDij ϕij • Modified test functions: ψmod

ij =1

lVkmϕij

AMFD = D1 AN D2, where

D1 = diag((lVkm)−1)D2 = diag(lDij )




𝑙𝑗𝑙𝐷


−𝑙𝑖𝑙𝐷


𝑙𝑗𝑙𝐷


−𝑙𝑖𝑙𝐷


2𝑙𝑖𝑗𝐷


𝛼 𝛽

𝑥𝑖𝐷

𝑥𝑙𝐷

𝑥𝑗𝐷

1


1


1


1


2


𝛼 𝛽



u(x) =∑(i,j)

DOFNij (u)ϕij =

∑(i,j)

(∫ xDj

xDi

u · eDij

)ϕij

=∑(i,j)


midpoint rule

ϕij =∑(i,j)



ij =1

lVkmϕij






𝑙𝑗𝑙𝐷


−𝑙𝑖𝑙𝐷


𝑙𝑗𝑙𝐷


−𝑙𝑖𝑙𝐷


2𝑙𝑖𝑗𝐷


𝛼 𝛽

𝑥𝑖𝐷

𝑥𝑙𝐷

𝑥𝑗𝐷

1


1


1


1


2


𝛼 𝛽



u(x) =∑(i,j)

DOFNij (u)ϕij =

∑(i,j)

(∫ xDj

xDi

u · eDij

)ϕij

=∑(i,j)


midpoint rule

ϕij =∑(i,j)



ij =1

lVkmϕij






𝑙𝑗𝑙𝐷


−𝑙𝑖𝑙𝐷


𝑙𝑗𝑙𝐷


−𝑙𝑖𝑙𝐷


2𝑙𝑖𝑗𝐷


𝛼 𝛽

𝑥𝑖𝐷

𝑥𝑙𝐷

𝑥𝑗𝐷

1


1


1


1


2


𝛼 𝛽



u(x) =∑(i,j)

DOFNij (u)ϕij =

∑(i,j)

(∫ xDj

xDi

u · eDij

)ϕij

=∑(i,j)


midpoint rule

ϕij

=∑(i,j)



ij =1

lVkmϕij






𝑙𝑗𝑙𝐷


−𝑙𝑖𝑙𝐷


𝑙𝑗𝑙𝐷


−𝑙𝑖𝑙𝐷


2𝑙𝑖𝑗𝐷


𝛼 𝛽

𝑥𝑖𝐷

𝑥𝑙𝐷

𝑥𝑗𝐷

1


1


1


1


2


𝛼 𝛽



u(x) =∑(i,j)

DOFNij (u)ϕij =

∑(i,j)

(∫ xDj

xDi

u · eDij

)ϕij

=∑(i,j)


midpoint rule

ϕij =∑(i,j)



ij =1

lVkmϕij






𝑙𝑗𝑙𝐷


−𝑙𝑖𝑙𝐷


𝑙𝑗𝑙𝐷


−𝑙𝑖𝑙𝐷


2𝑙𝑖𝑗𝐷


𝛼 𝛽

𝑥𝑖𝐷

𝑥𝑙𝐷

𝑥𝑗𝐷

1


1


1


1


2


𝛼 𝛽



u(x) =∑(i,j)

DOFNij (u)ϕij =

∑(i,j)

(∫ xDj

xDi

u · eDij

)ϕij

=∑(i,j)


midpoint rule

ϕij =∑(i,j)


• Modified basis functions: ϕmodij = lDij ϕij

• Modified test functions: ψmodij =

1

lVkmϕij






𝑙𝑗𝑙𝐷


−𝑙𝑖𝑙𝐷


𝑙𝑗𝑙𝐷


−𝑙𝑖𝑙𝐷


2𝑙𝑖𝑗𝐷


𝛼 𝛽

𝑥𝑖𝐷

𝑥𝑙𝐷

𝑥𝑗𝐷

𝑙𝑗𝑙𝐷


𝑙𝑖𝑙𝐷


𝑙𝑗𝑙𝐷


𝑙𝑖𝑙𝐷


2𝑙𝑖𝑗𝐷


𝛼 𝛽

Mimetic FDM Modified Nedelec FEM


u(x) =∑(i,j)

DOFNij (u)ϕij =

∑(i,j)

(∫ xDj

xDi

u · eDij

)ϕij

=∑(i,j)


midpoint rule

ϕij =∑(i,j)


• Modified basis functions: ϕmodij = lDij ϕij

• Modified test functions: ψmodij =

1

lVkmϕij






𝑙𝑗𝑙𝐷


−𝑙𝑖𝑙𝐷


𝑙𝑗𝑙𝐷


−𝑙𝑖𝑙𝐷


2𝑙𝑖𝑗𝐷


𝛼 𝛽

𝑥𝑖𝐷

𝑥𝑙𝐷

𝑥𝑗𝐷

𝑙𝑗𝑙𝐷


𝑙𝑖𝑙𝐷


𝑙𝑗𝑙𝐷


𝑙𝑖𝑙𝐷


2𝑙𝑖𝑗𝐷


𝛼 𝛽



u(x) =∑(i,j)

DOFNij (u)ϕij =

∑(i,j)

(∫ xDj

xDi

u · eDij

)ϕij

=∑(i,j)


midpoint rule

ϕij =∑(i,j)



ij =1

lVkmϕij






𝑙𝑗𝑙𝐷


−𝑙𝑖𝑙𝐷


𝑙𝑗𝑙𝐷


−𝑙𝑖𝑙𝐷


2𝑙𝑖𝑗𝐷


𝛼 𝛽

𝑥𝑖𝐷

𝑥𝑙𝐷

𝑥𝑗𝐷

𝑙𝑗𝑙𝐷


−𝑙𝑖𝑙𝐷


𝑙𝑗𝑙𝐷


−𝑙𝑖𝑙𝐷


2𝑙𝑖𝑗𝐷


𝛼 𝛽

𝑥𝑖𝐷

𝑥𝑙𝐷

𝑥𝑗𝐷



u(x) =∑(i,j)

DOFNij (u)ϕij =

∑(i,j)

(∫ xDj

xDi

u · eDij

)ϕij

=∑(i,j)


midpoint rule

ϕij =∑(i,j)



ij =1

lVkmϕij






𝑙𝑗𝑙𝐷


−𝑙𝑖𝑙𝐷


𝑙𝑗𝑙𝐷


−𝑙𝑖𝑙𝐷


2𝑙𝑖𝑗𝐷


𝛼 𝛽

𝑥𝑖𝐷

𝑥𝑙𝐷

𝑥𝑗𝐷

𝑙𝑗𝑙𝐷


−𝑙𝑖𝑙𝐷


𝑙𝑗𝑙𝐷


−𝑙𝑖𝑙𝐷


2𝑙𝑖𝑗𝐷


𝛼 𝛽

𝑥𝑖𝐷

𝑥𝑙𝐷

𝑥𝑗𝐷



u(x) =∑(i,j)

DOFNij (u)ϕij =

∑(i,j)

(∫ xDj

xDi

u · eDij

)ϕij

=∑(i,j)


midpoint rule

ϕij =∑(i,j)



ij =1

lVkmϕij




Relation Between Finite Element and Mimetic Finite Difference gradh divh

Raviart-Thomas FEM

4/3 2/3 2/3

2/3 2/3

2/3 2/3

2/3

2/3 2/32/3

RT basis functions on hexagons

Find ϕkm ∈ V RT (H) (dimV RT (H) = 12) s.t.‖ ϕkm ‖2

∗→ min, ‖ ϕ ‖∗'‖ ϕ ‖L2 , divϕkm = c∫ϕkm · njl = 0, ∀jl ∈ ∂H, jl 6= km∫ϕkm · nkm = 1

What is c?

c = divϕkm =1

|H|

∫H

divϕkm

=1

|H|

∫∂H

ϕkm · njl︸︷︷︸q±1

= ± 1

|H|

(Ref: Kuznetsov, & Repin 2003; Boiarkine, Kuznetsov, & Svyatskiy 2007; Pasciak & Vassilevski, SISC, 2008)



Raviart-Thomas FEM

𝐻

𝑘

𝑚




What is c?

c = divϕkm =1

|H|

∫H

divϕkm

=1

|H|

∫∂H


= ± 1

|H|




Raviart-Thomas FEM

𝐻

𝑘

𝑚




What is c?

c = divϕkm =1

|H|

∫H

divϕkm

=1

|H|

∫∂H


= ± 1

|H|




Raviart-Thomas FEM

𝐻

𝑘

𝑚




What is c?

c = divϕkm =1

|H|

∫H

divϕkm

=1

|H|

∫∂H


= ± 1

|H|




Mimetic FDM and Modified Raviart-Thomas FEM

Mimetic FD:

2𝑙𝑘𝑚𝑉

|𝐻|𝑙𝑖𝑗𝐷

−𝑙𝑘𝑙𝑉


−𝑙𝑘𝑙𝑉

|𝐻|𝑙𝑖𝑗𝐷 −𝑙𝑘𝑛

𝑉


−𝑙𝑘𝑛𝑉


−𝑙𝑘𝑚𝑉


−𝑙𝑘𝑚𝑉


𝑙𝑘𝑙𝑉


𝑙𝑘𝑛𝑉


𝑙𝑘𝑙𝑉


𝑙𝑘𝑛𝑉


𝑥𝑖𝐷 𝑥𝑗

𝐷

𝑥𝑛𝑉 𝑥𝑙

𝑉

𝑥𝑚𝑉

𝑥𝑘𝑉

Raviart-Thomas FEM:

2

|𝐻|

−1

|𝐻|

−1

|𝐻| −1

|𝐻|

−1

|𝐻|

−1

|𝐻|

−1

|𝐻|

1

|𝐻|

1

|𝐻|

1

|𝐻|

1

|𝐻|

Modified RT FEM:

• New basis functions:

ϕmodkm = lVkmϕkm

• New test functions:

ψmodkm =

1

lDijϕkm

AMFD = D1 ART D2

where

D1 = diag((lDij )−1)

D2 = diag(lVkm)



Mimetic FDM and Modified Raviart-Thomas FEM

Mimetic FD:

2𝑙𝑘𝑚𝑉


−𝑙𝑘𝑙𝑉


−𝑙𝑘𝑙𝑉

|𝐻|𝑙𝑖𝑗𝐷 −𝑙𝑘𝑛

𝑉


−𝑙𝑘𝑛𝑉


−𝑙𝑘𝑚𝑉


−𝑙𝑘𝑚𝑉


𝑙𝑘𝑙𝑉


𝑙𝑘𝑛𝑉


𝑙𝑘𝑙𝑉


𝑙𝑘𝑛𝑉


𝑥𝑖𝐷 𝑥𝑗

𝐷

𝑥𝑛𝑉 𝑥𝑙

𝑉

𝑥𝑚𝑉

𝑥𝑘𝑉

Raviart-Thomas FEM:

2

|𝐻|

−1

|𝐻|

−1

|𝐻| −1

|𝐻|

−1

|𝐻|

−1

|𝐻|

−1

|𝐻|

1

|𝐻|

1

|𝐻|

1

|𝐻|

1

|𝐻|

Modified RT FEM:

• New basis functions:

ϕmodkm = lVkmϕkm

• New test functions:

ψmodkm =

1

lDijϕkm

AMFD = D1 ART D2

where

D1 = diag((lDij )−1)

D2 = diag(lVkm)


Relation Between Finite Element and Mimetic Finite Difference Convergence

Error Analysis of Mimetic FDM based on FEM

curlh roth:

ANUN = bN =⇒ D1AND2︸︷︷︸

AMFD

D−12 UN︸︷︷︸UMFD

= D1bN︸︷︷︸

bMFD

=⇒ UMFD = D−12 UN

Therefore, we have

uMFDh (x) =

∑(i,j)

uMFDij ϕmod

ij (x) =∑(i,j)

1

lDijuNij l

Dij ϕij =

∑(i,j)

uNij ϕij = uN(x)

Base on the standard error analysis for the Nedelec FEM, we automatically have

‖u− uMFDh ‖rot ≤ Ch

gradh divh:

Based on the standard error analysis for the RT FEM, we automatically have

‖u− uMFDh ‖div ≤ Ch

Remarks:

• Assume sufficiently smooth solution u(x) and regular domain Ω• Proper discretization for f(x) and mass lumping for the FEM (Brezzi, Fortin, &

Marini 2006)




curlh roth:

ANUN = bN

=⇒ D1AND2︸︷︷︸

AMFD


= D1bN︸︷︷︸

bMFD


Therefore, we have

uMFDh (x) =

∑(i,j)

uMFDij ϕmod

ij (x) =∑(i,j)

1

lDijuNij l

Dij ϕij =

∑(i,j)

uNij ϕij = uN(x)



gradh divh:



Remarks:


Marini 2006)




curlh roth:


AMFD


= D1bN︸︷︷︸

bMFD


Therefore, we have

uMFDh (x) =

∑(i,j)

uMFDij ϕmod

ij (x) =∑(i,j)

1

lDijuNij l

Dij ϕij =

∑(i,j)

uNij ϕij = uN(x)



gradh divh:



Remarks:


Marini 2006)




curlh roth:


AMFD


= D1bN︸︷︷︸

bMFD


Therefore, we have

uMFDh (x) =

∑(i,j)

uMFDij ϕmod

ij (x) =∑(i,j)

1

lDijuNij l

Dij ϕij =

∑(i,j)

uNij ϕij = uN(x)



gradh divh:



Remarks:


Marini 2006)




curlh roth:


AMFD


= D1bN︸︷︷︸

bMFD


Therefore, we have

uMFDh (x) =

∑(i,j)

uMFDij ϕmod

ij (x) =∑(i,j)

1

lDijuNij l

Dij ϕij =

∑(i,j)

uNij ϕij = uN(x)



gradh divh:



Remarks:


Marini 2006)




curlh roth:


AMFD


= D1bN︸︷︷︸

bMFD


Therefore, we have

uMFDh (x) =

∑(i,j)

uMFDij ϕmod

ij (x) =∑(i,j)

1

lDijuNij l

Dij ϕij =

∑(i,j)

uNij ϕij = uN(x)



gradh divh:



Remarks:


Marini 2006)




curlh roth:


AMFD


= D1bN︸︷︷︸

bMFD


Therefore, we have

uMFDh (x) =

∑(i,j)

uMFDij ϕmod

ij (x) =∑(i,j)

1

lDijuNij l

Dij ϕij =

∑(i,j)

uNij ϕij = uN(x)



gradh divh:



Remarks:


Marini 2006)




curlh roth:


AMFD


= D1bN︸︷︷︸

bMFD


Therefore, we have

uMFDh (x) =

∑(i,j)

uMFDij ϕmod

ij (x) =∑(i,j)

1

lDijuNij l

Dij ϕij =

∑(i,j)

uNij ϕij = uN(x)



gradh divh:



Remarks:


Marini 2006)


Geometric Multigrid Methods

Outline

1 Introduction





Geometric Multigrid Methods Geometric Multigrid

Multigrid for Mimetic FDM: curlh roth

Approach: Use the relations between mimetic FDM and FEM

• Hierarchy of grids:

...

Components of

the vector grid

functions

• Choose components for GMG algorithm

• Smoothers• Intergrid transfer operators: prolongation and restriction• Coarse grid problems






...

Components of

the vector grid

functions








...

Components of

the vector grid

functions








...

Components of

the vector grid

functions





Schwarz-type Smoother

Multiplicative/Additive Schwarz-type smoothers

• Simultaneously update all the unknownsaround a vertex

• Solve 6× 6 systems of equations

• Overlapping among the blocks

(Ref: Arnold, Falk, & Winther, Numer. Math. 2000)



Intergrid Transfer Operators

Construct prolongation & restriction from the Nedelec canonical prolongation Q

• Prolongation: represent coarse grid basis as linear combination of finegrid basis

ϕH,modij = lD,H

ij ϕHij = lD,H

ij

∑kl

qijklϕ

hkl =

∑kl

pijkl︷︸︸︷

lD,Hij qij

kl

1

lD,hkl

ϕh,modkl︷︸︸︷

lD,hkl ϕh

kl =:∑kl

pijklϕ

h,modkl

Therefore, we haveP = (D2,h)−1 Q (D2,H)

• Restriction: similarly, R = (D1,H)QT (D1,h)−1

1

1

1/2

−sin(𝛼 + 𝛽)

2 sin 𝛼

1/2

−sin(𝛼 + 𝛽)

2 sin 𝛼

sin(𝛼 + 𝛽)

2 sin 𝛽

sin(𝛼 + 𝛽)

2 sin 𝛽

1/4

1/4

1/8

cos 𝛼

8 cos(𝛼 + 𝛽)

1/8

cos 𝛼

8 cos(𝛼 + 𝛽)

−cos 𝛽

8 cos(𝛼 + 𝛽)

−cos 𝛽

8 cos(𝛼 + 𝛽)

Prolongation Restriction






ϕH,modij = lD,H

ij ϕHij = lD,H

ij

∑kl

qijklϕ

hkl =

∑kl

pijkl︷︸︸︷

lD,Hij qij

kl

1

lD,hkl


lD,hkl ϕh

kl =:∑kl

pijklϕ

h,modkl



1

1

1/2

−sin(𝛼 + 𝛽)

2 sin 𝛼

1/2

−sin(𝛼 + 𝛽)

2 sin 𝛼

sin(𝛼 + 𝛽)

2 sin 𝛽

sin(𝛼 + 𝛽)

2 sin 𝛽

1/4

1/4

1/8

cos 𝛼

8 cos(𝛼 + 𝛽)

1/8

cos 𝛼

8 cos(𝛼 + 𝛽)

−cos 𝛽

8 cos(𝛼 + 𝛽)

−cos 𝛽

8 cos(𝛼 + 𝛽)







ϕH,modij = lD,H

ij ϕHij

= lD,Hij

∑kl

qijklϕ

hkl =

∑kl

pijkl︷︸︸︷

lD,Hij qij

kl

1

lD,hkl


lD,hkl ϕh

kl =:∑kl

pijklϕ

h,modkl



1

1

1/2

−sin(𝛼 + 𝛽)

2 sin 𝛼

1/2

−sin(𝛼 + 𝛽)

2 sin 𝛼

sin(𝛼 + 𝛽)

2 sin 𝛽

sin(𝛼 + 𝛽)

2 sin 𝛽

1/4

1/4

1/8

cos 𝛼

8 cos(𝛼 + 𝛽)

1/8

cos 𝛼

8 cos(𝛼 + 𝛽)

−cos 𝛽

8 cos(𝛼 + 𝛽)

−cos 𝛽

8 cos(𝛼 + 𝛽)







ϕH,modij = lD,H

ij ϕHij = lD,H

ij

∑kl

qijklϕ

hkl

=∑kl

pijkl︷︸︸︷

lD,Hij qij

kl

1

lD,hkl


lD,hkl ϕh

kl =:∑kl

pijklϕ

h,modkl



1

1

1/2

−sin(𝛼 + 𝛽)

2 sin 𝛼

1/2

−sin(𝛼 + 𝛽)

2 sin 𝛼

sin(𝛼 + 𝛽)

2 sin 𝛽

sin(𝛼 + 𝛽)

2 sin 𝛽

1/4

1/4

1/8

cos 𝛼

8 cos(𝛼 + 𝛽)

1/8

cos 𝛼

8 cos(𝛼 + 𝛽)

−cos 𝛽

8 cos(𝛼 + 𝛽)

−cos 𝛽

8 cos(𝛼 + 𝛽)







ϕH,modij = lD,H

ij ϕHij = lD,H

ij

∑kl

qijklϕ

hkl =

∑kl

pijkl︷︸︸︷

lD,Hij qij

kl

1

lD,hkl


lD,hkl ϕh

kl =:∑kl

pijklϕ

h,modkl



1

1

1/2

−sin(𝛼 + 𝛽)

2 sin 𝛼

1/2

−sin(𝛼 + 𝛽)

2 sin 𝛼

sin(𝛼 + 𝛽)

2 sin 𝛽

sin(𝛼 + 𝛽)

2 sin 𝛽

1/4

1/4

1/8

cos 𝛼

8 cos(𝛼 + 𝛽)

1/8

cos 𝛼

8 cos(𝛼 + 𝛽)

−cos 𝛽

8 cos(𝛼 + 𝛽)

−cos 𝛽

8 cos(𝛼 + 𝛽)







ϕH,modij = lD,H

ij ϕHij = lD,H

ij

∑kl

qijklϕ

hkl =

∑kl

pijkl︷︸︸︷

lD,Hij qij

kl

1

lD,hkl


lD,hkl ϕh

kl =:∑kl

pijklϕ

h,modkl



1

1

1/2

−sin(𝛼 + 𝛽)

2 sin 𝛼

1/2

−sin(𝛼 + 𝛽)

2 sin 𝛼

sin(𝛼 + 𝛽)

2 sin 𝛽

sin(𝛼 + 𝛽)

2 sin 𝛽

1/4

1/4

1/8

cos 𝛼

8 cos(𝛼 + 𝛽)

1/8

cos 𝛼

8 cos(𝛼 + 𝛽)

−cos 𝛽

8 cos(𝛼 + 𝛽)

−cos 𝛽

8 cos(𝛼 + 𝛽)







ϕH,modij = lD,H

ij ϕHij = lD,H

ij

∑kl

qijklϕ

hkl =

∑kl

pijkl︷︸︸︷

lD,Hij qij

kl

1

lD,hkl


lD,hkl ϕh

kl =:∑kl

pijklϕ

h,modkl



1

1

1/2

−sin(𝛼 + 𝛽)

2 sin 𝛼

1/2

−sin(𝛼 + 𝛽)

2 sin 𝛼

sin(𝛼 + 𝛽)

2 sin 𝛽

sin(𝛼 + 𝛽)

2 sin 𝛽

1/4

1/4

1/8

cos 𝛼

8 cos(𝛼 + 𝛽)

1/8

cos 𝛼

8 cos(𝛼 + 𝛽)

−cos 𝛽

8 cos(𝛼 + 𝛽)

−cos 𝛽

8 cos(𝛼 + 𝛽)







ϕH,modij = lD,H

ij ϕHij = lD,H

ij

∑kl

qijklϕ

hkl =

∑kl

pijkl︷︸︸︷

lD,Hij qij

kl

1

lD,hkl


lD,hkl ϕh

kl =:∑kl

pijklϕ

h,modkl



1

1

1/2

−sin(𝛼 + 𝛽)

2 sin 𝛼

1/2

−sin(𝛼 + 𝛽)

2 sin 𝛼

sin(𝛼 + 𝛽)

2 sin 𝛽

sin(𝛼 + 𝛽)

2 sin 𝛽

1/4

1/4

1/8

cos 𝛼

8 cos(𝛼 + 𝛽)

1/8

cos 𝛼

8 cos(𝛼 + 𝛽)

−cos 𝛽

8 cos(𝛼 + 𝛽)

−cos 𝛽

8 cos(𝛼 + 𝛽)




Coarse-grid Opertors

Rediscretization on the coarse grids satisfies AFDH = RAFD

h P

AFDH = D1,HA

NHD2,H = D1,HQ

TANh QD2,H

= D1,HQTD−1

1,h

AMFD︷︸︸︷D1,hA

Nh D2,h D

−12,hQD2,H

=

R︷︸︸︷(D1,HQ

TD−11,h)AMFD

h

P︷︸︸︷(D−1

2,hQD2,H)

= RAMFDh P




Rediscretization on the coarse grids

satisfies AFDH = RAFD

h P

AFDH = D1,HA

NHD2,H = D1,HQ

TANh QD2,H

= D1,HQTD−1

1,h


Nh D2,h D

−12,hQD2,H

=

R︷︸︸︷(D1,HQ

TD−11,h)AMFD

h

P︷︸︸︷(D−1

2,hQD2,H)

= RAMFDh P





h P

AFDH = D1,HA

NHD2,H = D1,HQ

TANh QD2,H

= D1,HQTD−1

1,h


Nh D2,h D

−12,hQD2,H

=

R︷︸︸︷(D1,HQ

TD−11,h)AMFD

h

P︷︸︸︷(D−1

2,hQD2,H)

= RAMFDh P





h P

AFDH = D1,HA

NHD2,H

= D1,HQTAN

h QD2,H

= D1,HQTD−1

1,h


Nh D2,h D

−12,hQD2,H

=

R︷︸︸︷(D1,HQ

TD−11,h)AMFD

h

P︷︸︸︷(D−1

2,hQD2,H)

= RAMFDh P





h P

AFDH = D1,HA

NHD2,H = D1,HQ

TANh QD2,H

= D1,HQTD−1

1,h


Nh D2,h D

−12,hQD2,H

=

R︷︸︸︷(D1,HQ

TD−11,h)AMFD

h

P︷︸︸︷(D−1

2,hQD2,H)

= RAMFDh P





h P

AFDH = D1,HA

NHD2,H = D1,HQ

TANh QD2,H

= D1,HQTD−1

1,h


Nh D2,h D

−12,hQD2,H

=

R︷︸︸︷(D1,HQ

TD−11,h)AMFD

h

P︷︸︸︷(D−1

2,hQD2,H)

= RAMFDh P





h P

AFDH = D1,HA

NHD2,H = D1,HQ

TANh QD2,H

= D1,HQTD−1

1,h


Nh D2,h D

−12,hQD2,H

=

R︷︸︸︷(D1,HQ

TD−11,h)AMFD

h

P︷︸︸︷(D−1

2,hQD2,H)

= RAMFDh P





h P

AFDH = D1,HA

NHD2,H = D1,HQ

TANh QD2,H

= D1,HQTD−1

1,h


Nh D2,h D

−12,hQD2,H

=

R︷︸︸︷(D1,HQ

TD−11,h)AMFD

h

P︷︸︸︷(D−1

2,hQD2,H)

= RAMFDh P



Multigrid for Mimetic FDM: gradh divh

One difficulty: non-nested meshes

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

One possible solution: go back to the triangle

! !!

!!

!!

! !!

!!

!!

ϕmodkm = lVkmϕkm = lVkm

∑i

αiϕRTi =

∑i

αi lVkm ϕRT

i

• Standard MG for H(div)• Auxiliary space preconditioner based on the regular decomposition of

H(div)





0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


! !!

!!

!!

! !!

!!

!!


∑i

αiϕRTi =

∑i

αi lVkm ϕRT

i


H(div)





0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


! !!

!!

!!

! !!

!!

!!


∑i

αiϕRTi =

∑i

αi lVkm ϕRT

i


H(div)





0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


! !!

!!

!!

! !!

!!

!!


∑i

αiϕRTi =

∑i

αi lVkm ϕRT

i


H(div)



Numerical Experiments: curlh roth

W-cycle V-cycle

ν ρ2g ρWh ρ3g ρVh1 0.331 0.330 0.337 0.334

2 0.124 0.124 0.133 0.132

3 0.070 0.069 0.072 0.071

4 0.045 0.045 0.052 0.052

• Accurate predictions by LocalFourier Analysis (LFA)

• Optimal convergence of GMG

Three-grid convergence rate predicted by LFA(different α and β)

• Convergence factor deteriorates whensmall angles appear

• Possible to improve by using a relaxationparameter ω(α = β = 80o: ρV3g = 0.508, butρV3g = 0.252 with ω = 1.35)

0.07

0.080.1

0.13

0.2

0.30.5

0.7

0.9

α

β

5 15 25 35 45 55 65 75 855

15

25

35

45

55

65

75

85



Numerical Experiments: curlh roth

W-cycle V-cycle

ν ρ2g ρWh ρ3g ρVh1 0.331 0.330 0.337 0.334

2 0.124 0.124 0.133 0.132

3 0.070 0.069 0.072 0.071

4 0.045 0.045 0.052 0.052

• Accurate predictions by LocalFourier Analysis (LFA)

• Optimal convergence of GMG

Three-grid convergence rate predicted by LFA(different α and β)

• Convergence factor deteriorates whensmall angles appear

• Possible to improve by using a relaxationparameter ω(α = β = 80o: ρV3g = 0.508, butρV3g = 0.252 with ω = 1.35)

0.07

0.080.1

0.13

0.2

0.30.5

0.7

0.9

α

β

5 15 25 35 45 55 65 75 855

15

25

35

45

55

65

75

85


Conclusions and Future Work

Outline

1 Introduction







Conclusions:

• Relation between Mimetic FDM and Petrov-Galerkin FEM

• Error Analysis for mimetic FDM can be derived from FEM framework

• Efficient GMG for curlh roth and gradh divh can be designed with the helpfrom FEM

Future Work:

• Other finite element families on polytopal meshes (Gillette, Rand, & Bajaj, 2014)

• Applications in different physical models: Darcy’s flow, Maxwell’sequation, etc




Conclusions:

• Relation between Mimetic FDM and Petrov-Galerkin FEM

• Error Analysis for mimetic FDM can be derived from FEM framework

• Efficient GMG for curlh roth and gradh divh can be designed with the helpfrom FEM

Future Work:

• Other finite element families on polytopal meshes (Gillette, Rand, & Bajaj, 2014)

• Applications in different physical models: Darcy’s flow, Maxwell’sequation, etc



Thank You!

Questions?
