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 Introduction 1 Relation Between Finite Element and Mimetic Finite Difference 2 Geometric Multigrid Methods 3 Conclusions and Future Work 4 X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 2 / 25
Introduction Outline Introduction 1 Relation Between Finite Element and Mimetic Finite Difference 2 Geometric Multigrid Methods 3 Conclusions and Future Work 4 X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 3 / 25
Introduction Model Problems: Model Equations curl rot u + κ u = f , in Ω − grad div u + κ u = f , in Ω X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 4 / 25
Introduction Model Problems: Model Equations curl rot u + κ u = f , in Ω − grad div u + κ u = 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. X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 4 / 25
Introduction Model Problems: Model Equations curl rot u + κ u = f , in Ω − grad div u + κ u = 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!! X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 4 / 25
Introduction Model Problems: Model Equations curl rot u + κ u = f , in Ω − grad div u + κ u = 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; Beir˜ ao 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 (Beir˜ ao 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 X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 5 / 25
Introduction Motivation A question: How to solve Ax = b efficiently This talk: 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) 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 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 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, Gallou¨ et, & Herbin 2010) X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 5 / 25
Introduction Mimetic FDM: Delaunay and Voronoi Grids Acute Delaunay grid Computational domain: Ω = Ω ∪ ∂ Ω { x D i , i = 1 , . . . , N D } X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 6 / 25
Introduction Mimetic FDM: Delaunay and Voronoi Grids Acute Delaunay grid Computational domain: Ω = Ω ∪ ∂ Ω { x D i , i = 1 , . . . , N D } Dual mesh: Voronoi grid Voronoi points: centers of the circumscribed circles on each triangle { x V k , i = 1 , . . . , N V } X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 6 / 25
Introduction Mimetic FDM: Delaunay and Voronoi Grids Acute Delaunay grid Computational domain: Ω = Ω ∪ ∂ Ω { x D i , i = 1 , . . . , N D } Dual mesh: Voronoi grid � D Voronoi points: centers of the circumscribed � circles on each triangle �� D � { x V k , i = 1 , . . . , N V } � For each x D Voronoi polygon: i V i = { x ∈ Ω | | x − x D i | < | x − x D j | , j = 1 , . . . , N D , j � = i } , and we denote: ∂ V ij = ∂ V i ∩ ∂ V j X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 6 / 25
Introduction Mimetic FDM: Grid Functions 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 ( x D i ) = u D at the nodes i x D i . H D denotes the set of u ( x ). • Voronoi grid: u ( x ) are defined by u ( x V k ) = u V k at the nodes x V k . H V denotes the set of u ( x ). 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 ( x D i ) = u D at the nodes i x D i . H D denotes the set of u ( x ). • Voronoi grid: u ( x ) are defined by u ( x V k ) = u V k at the nodes x V k . H V denotes the set of u ( x ). • Vector Grid Functions: • Delaunay grid: u ( x ) are defined by u ( x ) · e D ij = u D ij at the ij = 1 middle point of the edges x D 2 ( x D i + x D j ). H D denotes the set of u ( x ) • Voronoi grid: u ( x ) are defined by u ( x ) · e V km = u V km at the intersect points. H V denotes the set of u ( x ) e D ij 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
D � D �� V D � �� � D D D � � �� D � V � V V �� � D � D � �� � D � Introduction Mimetic FDM: Discrete Operators X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 8 / 25
D � V � V V �� � D � D � �� � D � Introduction Mimetic FDM: Discrete Operators Discrete Gradient Operators: grad h : H D → H D � 1 , ij := η ( i , j ) u D j − u D if j > i ( grad h u ) D i , with η ( i , j ) = D � l D − 1 , if j < i D ij �� V D � �� � D D D � � �� X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 8 / 25
D � V � V V �� � D � D � �� � D � Introduction Mimetic FDM: Discrete Operators Discrete Gradient Operators: grad h : H D → H D � 1 , ij := η ( i , j ) u D j − u D if j > i ( grad h u ) D i , with η ( i , j ) = D � l D − 1 , if j < i D ij �� V D � �� Discrete Rotor Operator: rot h : H D → H V � D D D � � �� k = η ( i , j ) u D ij l D ij + η ( j , l ) u D jl l D jl + η ( l , i ) u D li l D li ( rot h u ) V meas( D k ) X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 8 / 25
D � �� � D � Introduction Mimetic FDM: Discrete Operators Discrete Gradient Operators: grad h : H D → H D � 1 , ij := η ( i , j ) u D j − u D if j > i ( grad h u ) D i , with η ( i , j ) = D � l D − 1 , if j < i D ij �� V D � �� Discrete Rotor Operator: rot h : H D → H V � D D D � � �� k = η ( i , j ) u D ij l D ij + η ( j , l ) u D jl l D jl + η ( l , i ) u D li l D li ( rot h u ) V meas( D k ) D � V � V Discrete Curl Operator: curl h : H V → H D V �� � ij = η ( k , m ) u V k − u V D � ( curl h u ) D m l V km X. Hu (Tufts) Multigrid for Mimetic FDM Oct. 28, 2015 8 / 25
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