Finite Element Multigrid Framework for Mimetic …...Finite Element Multigrid Framework for Mimetic Finite Di erence Discretizations Xiaozhe Hu Tufts University Polytopal Element Methods
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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
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; 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; ...)
• ...
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; 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; ...)
• ...
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; 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; ...)
• ...
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; 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; ...)
• ...
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, Gallouet, & 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, Gallouet, & 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, Gallouet, & 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, Gallouet, & 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, Gallouet, & 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, Gallouet, & Herbin 2010)
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 5 / 25
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
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 6 / 25
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
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 6 / 25
Introduction
Mimetic FDM: Delaunay and Voronoi Grids
Computational domain:
Ω = Ω ∪ ∂Ω
D
D
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
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 6 / 25
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
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 7 / 25
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
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 7 / 25
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
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 7 / 25
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
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 8 / 25
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
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 8 / 25
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
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 8 / 25
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
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 8 / 25
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
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 8 / 25
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
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 9 / 25
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
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 9 / 25
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
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 9 / 25
Relation Between Finite Element and Mimetic Finite Difference
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 10 / 25
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
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 11 / 25
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
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 11 / 25
Relation Between Finite Element and Mimetic Finite Difference curlh roth
Mimetic FDM and Modified Nedelec FEM• General triangle:
𝑙𝑗𝑙𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
−𝑙𝑖𝑙𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
𝑙𝑗𝑙𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
−𝑙𝑖𝑙𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
2𝑙𝑖𝑗𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
𝛼 𝛽
𝑥𝑖𝐷
𝑥𝑙𝐷
𝑥𝑗𝐷
1
𝑚𝑒𝑎𝑠(𝐷𝑘) −
1
𝑚𝑒𝑎𝑠(𝐷𝑘)
1
𝑚𝑒𝑎𝑠(𝐷𝑘) −
1
𝑚𝑒𝑎𝑠(𝐷𝑘)
2
𝑚𝑒𝑎𝑠(𝐷𝑘)
𝛼 𝛽
Mimetic FDM Nedelec FEM
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 )
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 12 / 25
Relation Between Finite Element and Mimetic Finite Difference curlh roth
Mimetic FDM and Modified Nedelec FEM• General triangle:
𝑙𝑗𝑙𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
−𝑙𝑖𝑙𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
𝑙𝑗𝑙𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
−𝑙𝑖𝑙𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
2𝑙𝑖𝑗𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
𝛼 𝛽
𝑥𝑖𝐷
𝑥𝑙𝐷
𝑥𝑗𝐷
1
𝑚𝑒𝑎𝑠(𝐷𝑘) −
1
𝑚𝑒𝑎𝑠(𝐷𝑘)
1
𝑚𝑒𝑎𝑠(𝐷𝑘) −
1
𝑚𝑒𝑎𝑠(𝐷𝑘)
2
𝑚𝑒𝑎𝑠(𝐷𝑘)
𝛼 𝛽
Mimetic FDM Nedelec FEM
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 )
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 12 / 25
Relation Between Finite Element and Mimetic Finite Difference curlh roth
Mimetic FDM and Modified Nedelec FEM• General triangle:
𝑙𝑗𝑙𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
−𝑙𝑖𝑙𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
𝑙𝑗𝑙𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
−𝑙𝑖𝑙𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
2𝑙𝑖𝑗𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
𝛼 𝛽
𝑥𝑖𝐷
𝑥𝑙𝐷
𝑥𝑗𝐷
1
𝑚𝑒𝑎𝑠(𝐷𝑘) −
1
𝑚𝑒𝑎𝑠(𝐷𝑘)
1
𝑚𝑒𝑎𝑠(𝐷𝑘) −
1
𝑚𝑒𝑎𝑠(𝐷𝑘)
2
𝑚𝑒𝑎𝑠(𝐷𝑘)
𝛼 𝛽
Mimetic FDM Nedelec FEM
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 )
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 12 / 25
Relation Between Finite Element and Mimetic Finite Difference curlh roth
Mimetic FDM and Modified Nedelec FEM• General triangle:
𝑙𝑗𝑙𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
−𝑙𝑖𝑙𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
𝑙𝑗𝑙𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
−𝑙𝑖𝑙𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
2𝑙𝑖𝑗𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
𝛼 𝛽
𝑥𝑖𝐷
𝑥𝑙𝐷
𝑥𝑗𝐷
1
𝑚𝑒𝑎𝑠(𝐷𝑘) −
1
𝑚𝑒𝑎𝑠(𝐷𝑘)
1
𝑚𝑒𝑎𝑠(𝐷𝑘) −
1
𝑚𝑒𝑎𝑠(𝐷𝑘)
2
𝑚𝑒𝑎𝑠(𝐷𝑘)
𝛼 𝛽
Mimetic FDM Nedelec FEM
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 )
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 12 / 25
Relation Between Finite Element and Mimetic Finite Difference curlh roth
Mimetic FDM and Modified Nedelec FEM• General triangle:
𝑙𝑗𝑙𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
−𝑙𝑖𝑙𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
𝑙𝑗𝑙𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
−𝑙𝑖𝑙𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
2𝑙𝑖𝑗𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
𝛼 𝛽
𝑥𝑖𝐷
𝑥𝑙𝐷
𝑥𝑗𝐷
1
𝑚𝑒𝑎𝑠(𝐷𝑘) −
1
𝑚𝑒𝑎𝑠(𝐷𝑘)
1
𝑚𝑒𝑎𝑠(𝐷𝑘) −
1
𝑚𝑒𝑎𝑠(𝐷𝑘)
2
𝑚𝑒𝑎𝑠(𝐷𝑘)
𝛼 𝛽
Mimetic FDM Nedelec FEM
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 )
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 12 / 25
Relation Between Finite Element and Mimetic Finite Difference curlh roth
Mimetic FDM and Modified Nedelec FEM• General triangle:
𝑙𝑗𝑙𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
−𝑙𝑖𝑙𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
𝑙𝑗𝑙𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
−𝑙𝑖𝑙𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
2𝑙𝑖𝑗𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
𝛼 𝛽
𝑥𝑖𝐷
𝑥𝑙𝐷
𝑥𝑗𝐷
1
𝑚𝑒𝑎𝑠(𝐷𝑘) −
1
𝑚𝑒𝑎𝑠(𝐷𝑘)
1
𝑚𝑒𝑎𝑠(𝐷𝑘) −
1
𝑚𝑒𝑎𝑠(𝐷𝑘)
2
𝑚𝑒𝑎𝑠(𝐷𝑘)
𝛼 𝛽
Mimetic FDM Nedelec FEM
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: ψmodij =
1
lVkmϕij
AMFD = D1 AN D2, where
D1 = diag((lVkm)−1)D2 = diag(lDij )
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 12 / 25
Relation Between Finite Element and Mimetic Finite Difference curlh roth
Mimetic FDM and Modified Nedelec FEM• General triangle:
𝑙𝑗𝑙𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
−𝑙𝑖𝑙𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
𝑙𝑗𝑙𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
−𝑙𝑖𝑙𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
2𝑙𝑖𝑗𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
𝛼 𝛽
𝑥𝑖𝐷
𝑥𝑙𝐷
𝑥𝑗𝐷
𝑙𝑗𝑙𝐷
𝑚𝑒𝑎𝑠(𝐷𝑘) −
𝑙𝑖𝑙𝐷
𝑚𝑒𝑎𝑠(𝐷𝑘)
𝑙𝑗𝑙𝐷
𝑚𝑒𝑎𝑠(𝐷𝑘) −
𝑙𝑖𝑙𝐷
𝑚𝑒𝑎𝑠(𝐷𝑘)
2𝑙𝑖𝑗𝐷
𝑚𝑒𝑎𝑠(𝐷𝑘)
𝛼 𝛽
Mimetic FDM Modified Nedelec FEM
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: ψmodij =
1
lVkmϕij
AMFD = D1 AN D2, where
D1 = diag((lVkm)−1)D2 = diag(lDij )
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 12 / 25
Relation Between Finite Element and Mimetic Finite Difference curlh roth
Mimetic FDM and Modified Nedelec FEM• General triangle:
𝑙𝑗𝑙𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
−𝑙𝑖𝑙𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
𝑙𝑗𝑙𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
−𝑙𝑖𝑙𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
2𝑙𝑖𝑗𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
𝛼 𝛽
𝑥𝑖𝐷
𝑥𝑙𝐷
𝑥𝑗𝐷
𝑙𝑗𝑙𝐷
𝑚𝑒𝑎𝑠(𝐷𝑘) −
𝑙𝑖𝑙𝐷
𝑚𝑒𝑎𝑠(𝐷𝑘)
𝑙𝑗𝑙𝐷
𝑚𝑒𝑎𝑠(𝐷𝑘) −
𝑙𝑖𝑙𝐷
𝑚𝑒𝑎𝑠(𝐷𝑘)
2𝑙𝑖𝑗𝐷
𝑚𝑒𝑎𝑠(𝐷𝑘)
𝛼 𝛽
Mimetic FDM Modified Nedelec FEM
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 )
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 12 / 25
Relation Between Finite Element and Mimetic Finite Difference curlh roth
Mimetic FDM and Modified Nedelec FEM• General triangle:
𝑙𝑗𝑙𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
−𝑙𝑖𝑙𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
𝑙𝑗𝑙𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
−𝑙𝑖𝑙𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
2𝑙𝑖𝑗𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
𝛼 𝛽
𝑥𝑖𝐷
𝑥𝑙𝐷
𝑥𝑗𝐷
𝑙𝑗𝑙𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
−𝑙𝑖𝑙𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
𝑙𝑗𝑙𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
−𝑙𝑖𝑙𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
2𝑙𝑖𝑗𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
𝛼 𝛽
𝑥𝑖𝐷
𝑥𝑙𝐷
𝑥𝑗𝐷
Mimetic FDM Modified Nedelec FEM
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 )
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 12 / 25
Relation Between Finite Element and Mimetic Finite Difference curlh roth
Mimetic FDM and Modified Nedelec FEM• General triangle:
𝑙𝑗𝑙𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
−𝑙𝑖𝑙𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
𝑙𝑗𝑙𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
−𝑙𝑖𝑙𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
2𝑙𝑖𝑗𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
𝛼 𝛽
𝑥𝑖𝐷
𝑥𝑙𝐷
𝑥𝑗𝐷
𝑙𝑗𝑙𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
−𝑙𝑖𝑙𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
𝑙𝑗𝑙𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
−𝑙𝑖𝑙𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
2𝑙𝑖𝑗𝐷
𝑙𝑘𝑚𝑉 𝑚𝑒𝑎𝑠(𝐷𝑘)
𝛼 𝛽
𝑥𝑖𝐷
𝑥𝑙𝐷
𝑥𝑗𝐷
Mimetic FDM Modified Nedelec FEM
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 )
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 12 / 25
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)
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 13 / 25
Relation Between Finite Element and Mimetic Finite Difference gradh divh
Raviart-Thomas FEM
𝐻
𝑘
𝑚
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)
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 13 / 25
Relation Between Finite Element and Mimetic Finite Difference gradh divh
Raviart-Thomas FEM
𝐻
𝑘
𝑚
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)
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 13 / 25
Relation Between Finite Element and Mimetic Finite Difference gradh divh
Raviart-Thomas FEM
𝐻
𝑘
𝑚
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)
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 13 / 25
Relation Between Finite Element and Mimetic Finite Difference gradh divh
Mimetic FDM and Modified Raviart-Thomas FEM
Mimetic FD:
2𝑙𝑘𝑚𝑉
|𝐻|𝑙𝑖𝑗𝐷
−𝑙𝑘𝑙𝑉
|𝐻|𝑙𝑖𝑗𝐷
−𝑙𝑘𝑙𝑉
|𝐻|𝑙𝑖𝑗𝐷 −𝑙𝑘𝑛
𝑉
|𝐻|𝑙𝑖𝑗𝐷
−𝑙𝑘𝑛𝑉
|𝐻|𝑙𝑖𝑗𝐷
−𝑙𝑘𝑚𝑉
|𝐻|𝑙𝑖𝑗𝐷
−𝑙𝑘𝑚𝑉
|𝐻|𝑙𝑖𝑗𝐷
𝑙𝑘𝑙𝑉
|𝐻|𝑙𝑖𝑗𝐷
𝑙𝑘𝑛𝑉
|𝐻|𝑙𝑖𝑗𝐷
𝑙𝑘𝑙𝑉
|𝐻|𝑙𝑖𝑗𝐷
𝑙𝑘𝑛𝑉
|𝐻|𝑙𝑖𝑗𝐷
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)
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 14 / 25
Relation Between Finite Element and Mimetic Finite Difference gradh divh
Mimetic FDM and Modified Raviart-Thomas FEM
Mimetic FD:
2𝑙𝑘𝑚𝑉
|𝐻|𝑙𝑖𝑗𝐷
−𝑙𝑘𝑙𝑉
|𝐻|𝑙𝑖𝑗𝐷
−𝑙𝑘𝑙𝑉
|𝐻|𝑙𝑖𝑗𝐷 −𝑙𝑘𝑛
𝑉
|𝐻|𝑙𝑖𝑗𝐷
−𝑙𝑘𝑛𝑉
|𝐻|𝑙𝑖𝑗𝐷
−𝑙𝑘𝑚𝑉
|𝐻|𝑙𝑖𝑗𝐷
−𝑙𝑘𝑚𝑉
|𝐻|𝑙𝑖𝑗𝐷
𝑙𝑘𝑙𝑉
|𝐻|𝑙𝑖𝑗𝐷
𝑙𝑘𝑛𝑉
|𝐻|𝑙𝑖𝑗𝐷
𝑙𝑘𝑙𝑉
|𝐻|𝑙𝑖𝑗𝐷
𝑙𝑘𝑛𝑉
|𝐻|𝑙𝑖𝑗𝐷
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)
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 14 / 25
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)
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 15 / 25
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)
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 15 / 25
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)
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 15 / 25
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)
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 15 / 25
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)
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 15 / 25
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)
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 15 / 25
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)
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 15 / 25
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)
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 15 / 25
Geometric Multigrid Methods
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 16 / 25
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
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 17 / 25
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
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 17 / 25
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
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 17 / 25
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
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 17 / 25
Geometric Multigrid Methods Geometric Multigrid
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)
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 18 / 25
Geometric Multigrid Methods Geometric Multigrid
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
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 19 / 25
Geometric Multigrid Methods Geometric Multigrid
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
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 19 / 25
Geometric Multigrid Methods Geometric Multigrid
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,Hij
∑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
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 19 / 25
Geometric Multigrid Methods Geometric Multigrid
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
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 19 / 25
Geometric Multigrid Methods Geometric Multigrid
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
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 19 / 25
Geometric Multigrid Methods Geometric Multigrid
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
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 19 / 25
Geometric Multigrid Methods Geometric Multigrid
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
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 19 / 25
Geometric Multigrid Methods Geometric Multigrid
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
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 19 / 25
Geometric Multigrid Methods Geometric Multigrid
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
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 20 / 25
Geometric Multigrid Methods Geometric Multigrid
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
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 20 / 25
Geometric Multigrid Methods Geometric Multigrid
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
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 20 / 25
Geometric Multigrid Methods Geometric Multigrid
Coarse-grid Opertors
Rediscretization on the coarse grids satisfies AFDH = RAFD
h P
AFDH = D1,HA
NHD2,H
= D1,HQTAN
h 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
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 20 / 25
Geometric Multigrid Methods Geometric Multigrid
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
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 20 / 25
Geometric Multigrid Methods Geometric Multigrid
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
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 20 / 25
Geometric Multigrid Methods Geometric Multigrid
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
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 20 / 25
Geometric Multigrid Methods Geometric Multigrid
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
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 20 / 25
Geometric Multigrid Methods Geometric Multigrid
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)
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 21 / 25
Geometric Multigrid Methods Geometric Multigrid
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)
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 21 / 25
Geometric Multigrid Methods Geometric Multigrid
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)
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 21 / 25
Geometric Multigrid Methods Geometric Multigrid
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)
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 21 / 25
Geometric Multigrid Methods Geometric Multigrid
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
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 22 / 25
Geometric Multigrid Methods Geometric Multigrid
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
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 22 / 25
Conclusions and Future Work
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 23 / 25
Conclusions and Future Work
Conclusions and Future Work
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
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 24 / 25
Conclusions and Future Work
Conclusions and Future Work
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
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 24 / 25
Conclusions and Future Work
Thank You!
Questions?
X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 25 / 25
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