Finite Element Multigrid Framework for Mimetic Finite Difference Discretizations

Transcription

1 Finite Element Multigrid Framework for Mimetic Finite ifference iscretizations 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 FM Oct. 28, / 25

2 Outline 1 Introduction 2 Relation Between Finite Element and Mimetic Finite ifference 3 Geometric Multigrid Methods 4 Conclusions and Future Work X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

3 Outline Introduction 1 Introduction 2 Relation Between Finite Element and Mimetic Finite ifference 3 Geometric Multigrid Methods 4 Conclusions and Future Work X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

4 Introduction Model Problems: Model Equations curl rotu + κu = f, grad divu + κu = f, in Ω in Ω X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

5 Introduction Model Problems: Model Equations curl rotu + κu = f, grad divu + κu = f, in Ω in Ω Applications: arcy 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 FM Oct. 28, / 25

6 Introduction Model Problems: Model Equations curl rotu + κu = f, grad divu + κu = f, in Ω in Ω Applications: arcy 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 FM Oct. 28, / 25

7 Introduction Model Problems: Model Equations curl rotu + κu = f, grad divu + κu = f, in Ω in Ω Applications: arcy 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 a 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;...) iscontinuous Galerkin method (Arnold, Brezzi, Cockburn, & Marini 2002;...) Virtual element method (Beirão a Veiga, Brezzi, Cangiani, Manzini, Marini & Russo 2013;...) Weak Galerkin method (Wang & Ye 2013;...) Hybrid High-Order method (i Pietro, Ern, & Lemaire 2014;...)... X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

8 Introduction Motivation A question: How to solve Ax = b efficiently X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

9 Introduction Motivation A question: How to solve Ax = b efficiently This talk: X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

10 Introduction Motivation A question: How to solve Ax = b efficiently This talk: focus on mimetic FM (Vector Analysis Grid Operators Method, Vabishchevich, 2005) X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

11 Introduction Motivation A question: How to solve Ax = b efficiently This talk: focus on mimetic FM (Vector Analysis Grid Operators Method, Vabishchevich, 2005) show relation between mimetic FM and FEM X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

12 Introduction Motivation A question: How to solve Ax = b efficiently This talk: focus on mimetic FM (Vector Analysis Grid Operators Method, Vabishchevich, 2005) show relation between mimetic FM and FEM design geometric multigrid methods for mimetic FM X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

13 Introduction Motivation A question: How to solve Ax = b efficiently This talk: focus on mimetic FM (Vector Analysis Grid Operators Method, Vabishchevich, 2005) show relation between mimetic FM and FEM design geometric multigrid methods for mimetic FM Relation between MF and MFEM for diffusion (Berndt, Lipnikov, Moulton, & Shashkov 2001; Berndt, Lipnikov, Shashkov, Wheeler & Yotov 2005; roniou, Eymard, Gallouët, & Herbin 2010) X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

14 Introduction Mimetic FM: elaunay and Voronoi Grids Computational domain: Acute elaunay grid Ω = Ω Ω {x i, i = 1,..., N } X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

15 Introduction Mimetic FM: elaunay and Voronoi Grids Computational domain: Acute elaunay grid Ω = Ω Ω {x i, i = 1,..., N } ual 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 FM Oct. 28, / 25

16 Introduction Mimetic FM: elaunay and Voronoi Grids Computational domain: Acute elaunay grid Ω = Ω Ω {x i, i = 1,..., N } ual mesh: Voronoi grid Voronoi points: centers of the circumscribed circles on each triangle {x V k, i = 1,..., N V } For each x i Voronoi polygon: V i = {x Ω x x i < x x j, j = 1,..., N, j i}, and we denote: V = V i V j X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

17 Introduction Mimetic FM: Grid Functions X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

18 Introduction Mimetic FM: Grid Functions Scalar Grid Functions: elaunay grid: u(x) are defined by u(x i ) = ui x i. H denotes the set of u(x). Voronoi grid: u(x) are defined by u(x V k ) = uv k x V k. H V denotes the set of u(x). at the nodes at the nodes X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

19 Introduction Mimetic FM: Grid Functions Scalar Grid Functions: elaunay grid: u(x) are defined by u(x i ) = ui x i. H denotes the set of u(x). at the nodes Voronoi grid: u(x) are defined by u(x V k ) = uv k at the nodes x V k. H V denotes the set of u(x). Vector Grid Functions: elaunay grid: u(x) are defined by u(x) e = u at the middle point of the edges x = 1 2 (x i + x j ). H denotes the set of u(x) Voronoi grid: u(x) are defined by u(x) e V km = uv km at the intersect points. H V denotes the set of u(x) e is directed from the node with smaller index to the node with larger index X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

20 Introduction Mimetic FM: iscrete Operators X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

21 Introduction Mimetic FM: iscrete Operators iscrete Gradient Operators: grad h : H H (grad h u) :=η(i, j) u j u i l, with η(i, j) = { 1, if j > i 1, if j < i V X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

22 Introduction Mimetic FM: iscrete Operators iscrete Gradient Operators: grad h : H H (grad h u) :=η(i, j) u j u i l, with η(i, j) = { 1, if j > i 1, if j < i iscrete Rotor Operator: rot h : H H V (rot h u) V k = η(i, j) u l + η(j, l) ujl ljl meas( k ) + η(l, i) u li l li V X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

23 Introduction Mimetic FM: iscrete Operators iscrete Gradient Operators: grad h : H H (grad h u) :=η(i, j) u j u i l, with η(i, j) = { 1, if j > i 1, if j < i iscrete Rotor Operator: rot h : H H V (rot h u) V k = η(i, j) u l + η(j, l) ujl ljl meas( k ) + η(l, i) u li l li V V iscrete Curl Operator: curl h : H V H V V (curl h u) = η(k, m) uv k u V m l V km X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

24 Introduction Mimetic FM: iscrete Operators iscrete Gradient Operators: grad h : H H (grad h u) :=η(i, j) u j u i l, with η(i, j) = { 1, if j > i 1, if j < i iscrete Rotor Operator: rot h : H H V (rot h u) V k = η(i, j) u l + η(j, l) ujl ljl meas( k ) + η(l, i) u li l li V V iscrete Curl Operator: curl h : H V H V V (curl h u) = η(k, m) uv k u V m l V km iscrete ivergence Operator: div h : H H (div h u) i = 1 meas(v i ) j W V (i) u (e n V ) meas( V ) X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

25 Introduction Mimetic FM: Stencils 2/3 1/ 1/ 2/3 2/3 2/3 2/3 2/3 4 3/3 4 3/3 4 3/3 3/ 3/ grad h div h rot h curl h X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

26 Introduction Mimetic FM: Stencils 2/3 1/ 1/ 2/3 2/3 2/3 2/3 2/3 4 3/3 4 3/3 4 3/3 3/ 3/ grad h div h rot h curl h Mimetic FM curl h rot h u h + κu h = f h, grad h div h u h + κu h = f h, in Ω in Ω X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

27 Introduction Mimetic FM: Stencils 2/3 1/ 1/ 2/3 2/3 2/3 2/3 2/3 4 3/3 4 3/3 4 3/3 3/ 3/ grad h div h rot h curl h Mimetic FM curl h rot h u h + κu h = f h, grad h div h u h + κu h = f h, in Ω in Ω 4/ 8/ 4/ 2/3 2/3 2/3 2/3 2/3 4/3 2/3 4/ 4/ 2/3 2/3 2/3 2/3 curl h rot h grad h div h X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

28 Relation Between Finite Element and Mimetic Finite ifference Outline 1 Introduction 2 Relation Between Finite Element and Mimetic Finite ifference 3 Geometric Multigrid Methods 4 Conclusions and Future Work X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

29 Relation Between Finite Element and Mimetic Finite ifference Mimetic FM and Nédélec FEM curl h rot h Nédélec FEM Find u h V N h, such that, (rot u h, rot v h ) + κ(u h, v h ) = (f, v h ), where V N h consists lowest order Nédélec finite elements. v h V N h X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

30 Relation Between Finite Element and Mimetic Finite ifference Mimetic FM and Nédélec FEM curl h rot h Nédélec FEM Find u h V N h, such that, (rot u h, rot v h ) + κ(u h, v h ) = (f, v h ), where V N h consists lowest order Nédélec finite elements. v h V N h Equilateral triangle: Mimetic FM Nédélec FEM 4/h 2 4/h 2 4 3/3h 2 4 3/3h 2 8/h 2 8 3/3h 2 4/h 2 4/h 2 4 3/3h 2 4 3/3h 2 X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

31 Relation Between Finite Element and Mimetic Finite ifference Mimetic FM and Modified Nédélec FEM General triangle: l il l V km meas( k ) x i α x l 2l l V km meas( k ) l jl l V km meas( k ) β x j 1 meas( k ) α 2 meas( k ) curl h rot h 1 meas( k ) β l jl l V km meas( k ) l il l V km meas( k ) 1 meas( k ) 1 meas( k ) Mimetic FM Nédélec FEM X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

32 Relation Between Finite Element and Mimetic Finite ifference Mimetic FM and Modified Nédélec FEM General triangle: l il l V km meas( k ) x i α x l 2l l V km meas( k ) l jl l V km meas( k ) β x j 1 meas( k ) α 2 meas( k ) curl h rot h 1 meas( k ) β l jl l V km meas( k ) l il l V km meas( k ) 1 meas( k ) 1 meas( k ) Mimetic FM Consider a function u(x) V N h, Nédélec FEM X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

33 Relation Between Finite Element and Mimetic Finite ifference Mimetic FM and Modified Nédélec FEM General triangle: l il l V km meas( k ) x i α x l 2l l V km meas( k ) l jl l V km meas( k ) β x j 1 meas( k ) α 2 meas( k ) curl h rot h 1 meas( k ) β l jl l V km meas( k ) l il l V km meas( k ) 1 meas( k ) 1 meas( k ) Mimetic FM Consider a function u(x) V N h, u(x) = (i,j) OF N (u)ϕ = (i,j) ( x j x i Nédélec FEM ) u e ϕ X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

34 Relation Between Finite Element and Mimetic Finite ifference Mimetic FM and Modified Nédélec FEM General triangle: l il l V km meas( k ) x i α x l 2l l V km meas( k ) l jl l V km meas( k ) β x j 1 meas( k ) α 2 meas( k ) curl h rot h 1 meas( k ) β l jl l V km meas( k ) l il l V km meas( k ) 1 meas( k ) 1 meas( k ) Mimetic FM Consider a function u(x) V N h, u(x) = (i,j) OF N (u)ϕ = (i,j) ( x j x i Nédélec FEM ) u e ϕ = (u e )(x ) l }{{} (i,j) midpoint rule ϕ X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

35 Relation Between Finite Element and Mimetic Finite ifference Mimetic FM and Modified Nédélec FEM General triangle: l il l V km meas( k ) x i α x l 2l l V km meas( k ) l jl l V km meas( k ) β x j 1 meas( k ) α 2 meas( k ) curl h rot h 1 meas( k ) β l jl l V km meas( k ) l il l V km meas( k ) 1 meas( k ) 1 meas( k ) Mimetic FM Consider a function u(x) V N h, u(x) = (i,j) OF N (u)ϕ = (i,j) ( x j x i Nédélec FEM ) u e ϕ = (u e )(x ) l ϕ = }{{} (i,j) (i,j) midpoint rule OF MF (u)l ϕ X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

36 Relation Between Finite Element and Mimetic Finite ifference Mimetic FM and Modified Nédélec FEM General triangle: l il l V km meas( k ) x i α x l 2l l V km meas( k ) l jl l V km meas( k ) β x j 1 meas( k ) α 2 meas( k ) curl h rot h 1 meas( k ) β l jl l V km meas( k ) l il l V km meas( k ) 1 meas( k ) 1 meas( k ) Mimetic FM Consider a function u(x) V N h, u(x) = (i,j) OF N (u)ϕ = (i,j) ( x j x i Nédélec FEM ) u e ϕ Modified basis functions: ϕ mod = (u e )(x ) l ϕ = }{{} (i,j) (i,j) midpoint rule = l ϕ OF MF (u)l ϕ X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

37 Relation Between Finite Element and Mimetic Finite ifference Mimetic FM and Modified Nédélec FEM General triangle: l il l V km meas( k ) x i α x l 2l l V km meas( k ) l jl l V km meas( k ) β x j l il meas( k ) α 2l meas( k ) curl h rot h l jl meas( k ) β l jl l V km meas( k ) l il l V km meas( k ) l jl meas( k ) l il meas( k ) Mimetic FM Consider a function u(x) V N h, u(x) = (i,j) Modified basis functions: ϕ mod OF N (u)ϕ = (i,j) ( x j = (u e )(x ) l ϕ = }{{} (i,j) (i,j) midpoint rule = l ϕ Modified Nédélec FEM x i u e ) OF MF ϕ (u)l ϕ X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

38 Relation Between Finite Element and Mimetic Finite ifference Mimetic FM and Modified Nédélec FEM General triangle: l il l V km meas( k ) x i α x l 2l l V km meas( k ) l jl l V km meas( k ) β x j l il meas( k ) α 2l meas( k ) curl h rot h l jl meas( k ) β l jl l V km meas( k ) l il l V km meas( k ) l jl meas( k ) l il meas( k ) Mimetic FM Consider a function u(x) V N h, u(x) = (i,j) Modified basis functions: ϕ mod OF N (u)ϕ = (i,j) ( x j = (u e )(x ) l ϕ = }{{} (i,j) (i,j) midpoint rule Modified Nédélec FEM x i u e ) OF MF ϕ (u)l ϕ = l ϕ Modified test functions: ψ mod = 1 ϕ lkm V X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

39 Relation Between Finite Element and Mimetic Finite ifference Mimetic FM and Modified Nédélec FEM l il l V km meas( k ) x i General triangle: α x l 2l l V km meas( k ) l jl l V km meas( k ) β x j l il l V km meas( k ) x i α x l 2l l V km meas( k ) curl h rot h l jl l V km meas( k ) β x j l jl l V km meas( k ) l il l V km meas( k ) l jl l V km meas( k ) l il l V km meas( k ) Mimetic FM Consider a function u(x) V N h, u(x) = (i,j) Modified basis functions: ϕ mod OF N (u)ϕ = (i,j) ( x j = (u e )(x ) l ϕ = }{{} (i,j) (i,j) midpoint rule Modified Nédélec FEM x i u e ) OF MF ϕ (u)l ϕ = l ϕ Modified test functions: ψ mod = 1 ϕ lkm V X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

40 Relation Between Finite Element and Mimetic Finite ifference Mimetic FM and Modified Nédélec FEM l il l V km meas( k ) x i General triangle: α x l 2l l V km meas( k ) l jl l V km meas( k ) β x j l il l V km meas( k ) x i α x l 2l l V km meas( k ) curl h rot h l jl l V km meas( k ) β x j l jl l V km meas( k ) l il l V km meas( k ) l jl l V km meas( k ) l il l V km meas( k ) Mimetic FM Consider a function u(x) V N h, u(x) = (i,j) Modified basis functions: ϕ mod OF N (u)ϕ = (i,j) ( x j = (u e )(x ) l ϕ = }{{} (i,j) (i,j) midpoint rule Modified Nédélec FEM x i u e ) OF MF ϕ (u)l ϕ = l ϕ Modified test functions: ψ mod = 1 ϕ lkm V A MF = 1 A N 2, where { 1 = diag((l V km) 1 ) 2 = diag(l ) X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

41 Relation Between Finite Element and Mimetic Finite ifference Raviart-Thomas FEM grad h div h 2/3 2/3 2/3 2/3 2/3 4/3 2/3 2/3 2/3 2/3 2/3 X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

42 Relation Between Finite Element and Mimetic Finite ifference Raviart-Thomas FEM grad h div h H k m RT basis functions on hexagons X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

43 Relation Between Finite Element and Mimetic Finite ifference Raviart-Thomas FEM grad h div h H k m RT basis functions on hexagons Find ϕ km V RT (H) (dim V RT (H) = 12) s.t. ϕ km 2 min, ϕ ϕ L 2, div ϕ km = c ϕ km n jl = 0, jl H, jl km ϕ km n km = 1 (Ref: Kuznetsov, & Repin 2003; Boiarkine, Kuznetsov, & Svyatskiy 2007; Pasciak & Vassilevski, SISC, 2008) X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

44 Relation Between Finite Element and Mimetic Finite ifference Raviart-Thomas FEM grad h div h H k m RT basis functions on hexagons Find ϕ km V RT (H) (dim V RT (H) = 12) s.t. ϕ km 2 min, ϕ ϕ L 2, div ϕ km = c ϕ km n jl = 0, jl H, jl km ϕ km n km = 1 What is c? c = divϕ km = 1 H = 1 H ϕ km n jl H } {{ } ±1 H divϕ km = ± 1 H (Ref: Kuznetsov, & Repin 2003; Boiarkine, Kuznetsov, & Svyatskiy 2007; Pasciak & Vassilevski, SISC, 2008) X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

45 Relation Between Finite Element and Mimetic Finite ifference grad h div h Mimetic FM and Modified Raviart-Thomas FEM Mimetic F: V l km H l V l kl H l x n V V l kn H l V l kn H l x i V l kl H l x k V x m V V l kl H l V 2l km H l V l kn H l x l V x j V l kl H l V l kn H l V l km H l Raviart-Thomas FEM: 1 H 1 H 1 H 1 H 1 H 2 H 1 H 1 H 1 H 1 H 1 H X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

46 Relation Between Finite Element and Mimetic Finite ifference grad h div h Mimetic FM and Modified Raviart-Thomas FEM Mimetic F: V l km H l V l kl H l x n V V l kn H l V l kn H l x i V l kl H l x k V Raviart-Thomas FEM: x m V V l kl H l V 2l km H l V l kn H l x l V x j V l kl H l V l kn H l V l km H l Modified RT FEM: New basis functions: ϕ mod km = l V kmϕ km New test functions: ψ mod km = 1 l ϕ km 1 H 1 H 1 H 1 H A MF = 1 A RT 2 1 H 2 H 1 H where { 1 = diag((l )1 ) 2 = diag(l V km ) 1 H 1 H 1 H 1 H X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

47 Relation Between Finite Element and Mimetic Finite ifference Convergence Error Analysis of Mimetic FM based on FEM X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

48 Relation Between Finite Element and Mimetic Finite ifference Convergence Error Analysis of Mimetic FM based on FEM curl h rot h : A N U N = b N X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

49 Relation Between Finite Element and Mimetic Finite ifference Convergence Error Analysis of Mimetic FM based on FEM curl h rot h : A N U N = b N = 1A N }{{} 2 A MF 1 2 U N }{{} U MF N = }{{ 1b } b MF X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

50 Relation Between Finite Element and Mimetic Finite ifference Convergence Error Analysis of Mimetic FM based on FEM curl h rot h : A N U N = b N = 1A N }{{} 2 A MF 1 2 U N }{{} U MF N = }{{ 1b = U MF = 1 2 U N } b MF X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

51 Relation Between Finite Element and Mimetic Finite ifference Convergence Error Analysis of Mimetic FM based on FEM curl h rot h : A N U N = b N = 1A N }{{} 2 A MF Therefore, we have u MF h (x) = (i,j) u MF 1 2 U N }{{} U MF ϕ mod (x) = (i,j) N = }{{ 1b = U MF = 1 2 U N } b MF 1 l u N l ϕ = u N ϕ = u N (x) (i,j) X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

52 Relation Between Finite Element and Mimetic Finite ifference Convergence Error Analysis of Mimetic FM based on FEM curl h rot h : A N U N = b N = 1A N }{{} 2 A MF Therefore, we have u MF h (x) = (i,j) u MF 1 2 U N }{{} U MF ϕ mod (x) = (i,j) N = }{{ 1b = U MF = 1 2 U N } b MF 1 l u N l ϕ = u N ϕ = u N (x) (i,j) Base on the standard error analysis for the Nédélec FEM, we automatically have u u MF h rot Ch X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

53 Relation Between Finite Element and Mimetic Finite ifference Convergence Error Analysis of Mimetic FM based on FEM curl h rot h : A N U N = b N = 1A N }{{} 2 A MF Therefore, we have u MF h (x) = (i,j) u MF 1 2 U N }{{} U MF ϕ mod (x) = (i,j) N = }{{ 1b = U MF = 1 2 U N } b MF 1 l u N l ϕ = u N ϕ = u N (x) (i,j) Base on the standard error analysis for the Nédélec FEM, we automatically have grad h div h : u u MF h rot Ch Based on the standard error analysis for the RT FEM, we automatically have u u MF h div Ch X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

54 Relation Between Finite Element and Mimetic Finite ifference Convergence Error Analysis of Mimetic FM based on FEM curl h rot h : A N U N = b N = 1A N }{{} 2 A MF Therefore, we have u MF h (x) = (i,j) u MF 1 2 U N }{{} U MF ϕ mod (x) = (i,j) N = }{{ 1b = U MF = 1 2 U N } b MF 1 l u N l ϕ = u N ϕ = u N (x) (i,j) Base on the standard error analysis for the Nédélec FEM, we automatically have grad h div h : u u MF h rot Ch Based on the standard error analysis for the RT FEM, we automatically have Remarks: u u MF h div Ch 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 FM Oct. 28, / 25

55 Geometric Multigrid Methods Outline 1 Introduction 2 Relation Between Finite Element and Mimetic Finite ifference 3 Geometric Multigrid Methods 4 Conclusions and Future Work X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

56 Geometric Multigrid Methods Geometric Multigrid Multigrid for Mimetic FM: curl h rot h X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

57 Geometric Multigrid Methods Geometric Multigrid Multigrid for Mimetic FM: curl h rot h Approach: Use the relations between mimetic FM and FEM X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

58 Geometric Multigrid Methods Geometric Multigrid Multigrid for Mimetic FM: curl h rot h Approach: Use the relations between mimetic FM and FEM Hierarchy of grids: Components of the vector grid functions... X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

59 Geometric Multigrid Methods Geometric Multigrid Multigrid for Mimetic FM: curl h rot h Approach: Use the relations between mimetic FM 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 FM Oct. 28, / 25

60 Geometric Multigrid Methods Schwarz-type Smoother Geometric Multigrid Multiplicative/Additive Schwarz-type smoothers Simultaneously update all the unknowns around 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 FM Oct. 28, / 25

61 Geometric Multigrid Methods Intergrid Transfer Operators Geometric Multigrid X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

62 Geometric Multigrid Methods Intergrid Transfer Operators Geometric Multigrid Construct prolongation & restriction from the Nédélec canonical prolongation Q X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

63 Geometric Multigrid Methods Intergrid Transfer Operators Geometric Multigrid Construct prolongation & restriction from the Nédélec canonical prolongation Q Prolongation: represent coarse grid basis as linear combination of fine grid basis ϕ H,mod = l,h ϕ H X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

64 Geometric Multigrid Methods Intergrid Transfer Operators Geometric Multigrid Construct prolongation & restriction from the Nédélec canonical prolongation Q Prolongation: represent coarse grid basis as linear combination of fine grid basis ϕ H,mod = l,h ϕ H = l,h kl q kl ϕh kl X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

65 Geometric Multigrid Methods Intergrid Transfer Operators Geometric Multigrid Construct prolongation & restriction from the Nédélec canonical prolongation Q Prolongation: represent coarse grid basis as linear combination of fine grid basis p kl ϕ ϕ H,mod = l,h ϕ H = l,h q kl ϕh kl = {}}{ h,mod kl {}}{ l,h q 1 kl l,h l,h kl ϕ h kl =: kl kl kl kl p kl ϕh,mod kl X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

66 Geometric Multigrid Methods Intergrid Transfer Operators Geometric Multigrid Construct prolongation & restriction from the Nédélec canonical prolongation Q Prolongation: represent coarse grid basis as linear combination of fine grid basis p kl ϕ ϕ H,mod = l,h ϕ H = l,h q kl ϕh kl = {}}{ h,mod kl {}}{ l,h q 1 kl l,h l,h kl ϕ h kl =: kl kl kl kl Therefore, we have P = ( 2,h ) 1 Q ( 2,H ) p kl ϕh,mod kl X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

67 Geometric Multigrid Methods Intergrid Transfer Operators Geometric Multigrid Construct prolongation & restriction from the Nédélec canonical prolongation Q Prolongation: represent coarse grid basis as linear combination of fine grid basis p kl ϕ ϕ H,mod = l,h ϕ H = l,h q kl ϕh kl = {}}{ h,mod kl {}}{ l,h q 1 kl l,h l,h kl ϕ h kl =: kl kl kl kl Therefore, we have P = ( 2,h ) 1 Q ( 2,H ) Restriction: similarly, R = ( 1,H ) Q T ( 1,h ) 1 p kl ϕh,mod kl X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

68 Geometric Multigrid Methods Intergrid Transfer Operators Geometric Multigrid Construct prolongation & restriction from the Nédélec canonical prolongation Q Prolongation: represent coarse grid basis as linear combination of fine grid basis p kl ϕ ϕ H,mod = l,h ϕ H = l,h q kl ϕh kl = {}}{ h,mod kl {}}{ l,h q 1 kl l,h l,h kl ϕ h kl =: kl kl kl kl Therefore, we have P = ( 2,h ) 1 Q ( 2,H ) Restriction: similarly, R = ( 1,H ) Q T ( 1,h ) 1 p kl ϕh,mod kl sin(α + β) 2 sin α 1/2 sin(α + β) 2 sin β 1 cos α 8 cos(α + β) 1/8 cos β 8 cos(α + β) 1/4 1 sin(α + β) 2 sin α 1/4 cos α 8 cos(α + β) sin(α + β) 2 sin β 1/2 cos β 8 cos(α + β) 1/8 Prolongation Restriction X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

69 Geometric Multigrid Methods Coarse-grid Opertors Geometric Multigrid X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

70 Geometric Multigrid Methods Coarse-grid Opertors Geometric Multigrid Rediscretization on the coarse grids X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

71 Geometric Multigrid Methods Coarse-grid Opertors Geometric Multigrid Rediscretization on the coarse grids satisfies A F H = RAF h P X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

72 Geometric Multigrid Methods Coarse-grid Opertors Geometric Multigrid Rediscretization on the coarse grids satisfies A F H = RAF h P A F H = 1,H A N H 2,H X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

73 Geometric Multigrid Methods Coarse-grid Opertors Geometric Multigrid Rediscretization on the coarse grids satisfies A F H = RAF h P A F H = 1,H A N H 2,H = 1,H Q T A N h Q 2,H X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

74 Geometric Multigrid Methods Coarse-grid Opertors Geometric Multigrid Rediscretization on the coarse grids satisfies A F H = RAF h P A F H = 1,H A N H 2,H = 1,H Q T A N h Q 2,H = 1,H Q T 1 1,h A MF {}}{ 1,h A N h 2,h 1 2,h Q 2,H X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

75 Geometric Multigrid Methods Coarse-grid Opertors Geometric Multigrid Rediscretization on the coarse grids satisfies A F H = RAF h P A F H = 1,H A N H 2,H = 1,H Q T A N h Q 2,H = 1,H Q T 1 1,h = A MF {}}{ 1,h A N h 2,h 1 2,h Q 2,H R P {}}{{}}{ ( 1,H Q T 1 1,h ) AMF h ( 1 2,h Q 2,H) X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

76 Geometric Multigrid Methods Coarse-grid Opertors Geometric Multigrid Rediscretization on the coarse grids satisfies A F H = RAF h P A F H = 1,H A N H 2,H = 1,H Q T A N h Q 2,H = 1,H Q T 1 1,h A MF {}}{ 1,h A N h 2,h 1 2,h Q 2,H { R }} { { P }} { = ( 1,H Q T 1 1,h ) AMF h ( 1 2,h Q 2,H) = RA MF h P X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

77 Geometric Multigrid Methods Geometric Multigrid Multigrid for Mimetic FM: grad h div h One difficulty: non-nested meshes X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

78 Geometric Multigrid Methods Geometric Multigrid Multigrid for Mimetic FM: grad h div h One difficulty: non-nested meshes ! !! One possible solution: go back to the triangle!!!!!!!!!! X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

79 Geometric Multigrid Methods Geometric Multigrid Multigrid for Mimetic FM: grad h div h One difficulty: non-nested meshes ! !! One possible solution: go back to the triangle!!!!!! ϕ mod km = l V kmϕ km = l V km i!! α i ϕ RT i = i!! α i l V km ϕ RT i X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

80 Geometric Multigrid Methods Geometric Multigrid Multigrid for Mimetic FM: grad h div h One difficulty: non-nested meshes ! !! One possible solution: go back to the triangle!!!!!! ϕ mod km = l V kmϕ km = l V km i Standard MG for H(div) Auxiliary space preconditioner based on the regular decomposition of H(div)!! α i ϕ RT i = i!! α i l V km ϕ RT i X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

81 Geometric Multigrid Methods Geometric Multigrid Numerical Experiments: curl h rot h W-cycle V-cycle ν ρ 2g ρ W h ρ 3g ρ V h Accurate predictions by Local Fourier Analysis (LFA) Optimal convergence of GMG X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

82 Geometric Multigrid Methods Geometric Multigrid Numerical Experiments: curl h rot h W-cycle V-cycle ν ρ 2g ρ W h ρ 3g ρ V h Accurate predictions by Local Fourier Analysis (LFA) Optimal convergence of GMG Three-grid convergence rate predicted by LFA (different α and β) Convergence factor deteriorates when small angles appear Possible to improve by using a relaxation parameter ω (α = β = 80 o : ρ V 3g = 0.508, but ρ V 3g = with ω = 1.35) β α X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

83 Conclusions and Future Work Outline 1 Introduction 2 Relation Between Finite Element and Mimetic Finite ifference 3 Geometric Multigrid Methods 4 Conclusions and Future Work X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

84 Conclusions and Future Work Conclusions and Future Work Conclusions: Relation between Mimetic FM and Petrov-Galerkin FEM Error Analysis for mimetic FM can be derived from FEM framework Efficient GMG for curl h rot h and grad h div h can be designed with the help from FEM X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

85 Conclusions and Future Work Conclusions and Future Work Conclusions: Relation between Mimetic FM and Petrov-Galerkin FEM Error Analysis for mimetic FM can be derived from FEM framework Efficient GMG for curl h rot h and grad h div h can be designed with the help from FEM Future Work: Other finite element families on polytopal meshes (Gillette, Rand, & Bajaj, 2014) Applications in different physical models: arcy s flow, Maxwell s equation, etc X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25

86 Conclusions and Future Work Thank You! Questions? X. Hu (Tufts) Multigrid for Mimetic FM Oct. 28, / 25