Multigrid Methods for Maxwell s Equations

Transcription

1 Multigrid Methods for Maxwell s Equations Jintao Cui Institute for Mathematics and Its Applications University of Minnesota

2 Outline Nonconforming Finite Element Methods for a Two Dimensional Curl-Curl and Grad-Div Problem Maxwell s equations Curl-curl and grad-div problem (CCGD) Numerical schemes and error estimates Numerical results

3 Outline Nonconforming Finite Element Methods for a Two Dimensional Curl-Curl and Grad-Div Problem Maxwell s equations Curl-curl and grad-div problem (CCGD) Numerical schemes and error estimates Numerical results Multigrid Methods for Discontinuous Galerkin (DG) Methods on Graded Meshes Symmetric interior penalty (SIP) method and error estimates Multigrid algorithms Convergence analysis and numerical results

4 Outline Multigrid Methods for Maxwell s Equations Multigrid results Hodge decomposition and Maxwell s equations Concluding remarks

5 Nonconforming Finite Element Methods for a Two Dimensional Curl-Curl and Grad-Div Problem

6 Maxwell s Equations E = µ H t H = ɛ E t + J E = ρ ɛ H = 0 Faraday s law Ampère s law Gauss law-electric Gauss law-magnetic E : Electric field H : Magnetic field ρ: Charge density J: Current density ɛ: Electric permittivity µ: Magnetic permeability

7 Maxwell s Equations When the radiation has a frequency ω > 0 and charge is conserved, we want to find solutions of the Maxwell s equations of the form E(x, t) = e iωt Ê(x), H(x, t) = e iωt Ĥ(x), J(x, t) = e iωt Ĵ(x) Maxwell s equations become { (µh) + E = 0 t (ɛe) H = J t { { Ê ω2 µɛê Ĥ ω2 µɛĥ Ê = Ĥ = = iωµĵ = Ĵ iωµĥ iωɛê + Ĵ

8 Curl-Curl Problem We want to solve the equation of the following form with perfectly conducting boundary condition u + αu = f n u = 0 in Ω on Ω where Ω R 2 is a bounded polygonal domain.

9 Curl-Curl Problem We want to solve the equation of the following form with perfectly conducting boundary condition u + αu = f n u = 0 in Ω on Ω where Ω R 2 is a bounded polygonal domain. This problem appears in the semi-discretization of electric fields in the time-harmonic (frequency-domain) Maxwell s equations when α 0, and in the time-dependent (time-domain) Maxwell s equations when α > 0

10 Curl-Curl Problem Define H(curl; Ω) = {v [L 2 (Ω)] 2 : v L 2 (Ω)} H 0 (curl; Ω) = {v H(curl; Ω) : n v Ω = 0}

11 Curl-Curl Problem Define H(curl; Ω) = {v [L 2 (Ω)] 2 : v L 2 (Ω)} H 0 (curl; Ω) = {v H(curl; Ω) : n v Ω = 0} For any v H 0 (curl; Ω), ( u, v) L2 (Ω) + α(u, v) L2 (Ω) = (f, v) L2 (Ω) Weak form: Find u H 0 (curl; Ω) such that ( u, v) + α(u, v) = (f, v) v H 0 (curl; Ω) The disadvantage of working with this weak form is that it is non-elliptic.

12 Helmholtz decomposition For any u H 0 (curl; Ω), u = ů + ζ with ů H(div 0 ; Ω) and ζ H 1 0(Ω), where H(div 0 ; Ω) = {v [L 2 (Ω)] 2 : v = 0}.

13 Helmholtz decomposition For any u H 0 (curl; Ω), u = ů + ζ with ů H(div 0 ; Ω) and ζ H 1 0(Ω), where H(div 0 ; Ω) = {v [L 2 (Ω)] 2 : v = 0}. (Weak form) ( u, v) + α(u, v) = (f, v) Let η H0(Ω), 1 then v = η H 0 (curl; Ω), and ( u, ( η)) + α(u, η) =(f, η)

14 Helmholtz decomposition For any u H 0 (curl; Ω), u = ů + ζ with ů H(div 0 ; Ω) and ζ H 1 0(Ω), where H(div 0 ; Ω) = {v [L 2 (Ω)] 2 : v = 0}. (Weak form) ( u, v) + α(u, v) = (f, v) Let η H0(Ω), 1 then v = η H 0 (curl; Ω), and ( u, ( η)) + α(u, η) =(f, η) α(ů + ζ, η) =(f, η)

15 Helmholtz decomposition For any u H 0 (curl; Ω), u = ů + ζ with ů H(div 0 ; Ω) and ζ H 1 0(Ω), where H(div 0 ; Ω) = {v [L 2 (Ω)] 2 : v = 0}. (Weak form) ( u, v) + α(u, v) = (f, v) Let η H0(Ω), 1 then v = η H 0 (curl; Ω), and ( u, ( η)) + α(u, η) =(f, η) α(ů + ζ, η) =(f, η) α( ů, η) + α( ζ, η) =(f, η)

16 Helmholtz decomposition So ζ H 1 0(Ω) satisfies α( ζ, η) = (f, η) η H 1 0(Ω), which is the variational form of the Poisson problem.

17 Helmholtz decomposition So ζ H 1 0(Ω) satisfies α( ζ, η) = (f, η) η H 1 0(Ω), which is the variational form of the Poisson problem. (Weak form) ( u, v) + α(u, v) = (f, v) Let v = v H 0 (curl; Ω) H(div 0 ; Ω), ( u, v) + α(u, v) =(f, v)

18 Helmholtz decomposition So ζ H 1 0(Ω) satisfies α( ζ, η) = (f, η) η H 1 0(Ω), which is the variational form of the Poisson problem. (Weak form) ( u, v) + α(u, v) = (f, v) Let v = v H 0 (curl; Ω) H(div 0 ; Ω), ( u, v) + α(u, v) =(f, v) ( (ů + ζ), v) + α(ů + ζ, v) =(f, v)

19 Helmholtz decomposition So ζ H 1 0(Ω) satisfies α( ζ, η) = (f, η) η H 1 0(Ω), which is the variational form of the Poisson problem. (Weak form) ( u, v) + α(u, v) = (f, v) Let v = v H 0 (curl; Ω) H(div 0 ; Ω), ( u, v) + α(u, v) =(f, v) ( (ů + ζ), v) + α(ů + ζ, v) =(f, v) ( ů, v) + α(ů, v) =(f, v)

20 Reduced Curl-Curl Problem (RCCP) Hence ů satisfies the reduced curl-curl problem (RCCP) Find ů H 0 (curl; Ω) H(div 0 ; Ω) such that ( ů, v) + α(ů, v) = (f, v) v H 0 (curl; Ω) H(div 0 ; Ω)

21 Reduced Curl-Curl Problem (RCCP) Hence ů satisfies the reduced curl-curl problem (RCCP) Find ů H 0 (curl; Ω) H(div 0 ; Ω) such that ( ů, v) + α(ů, v) = (f, v) v H 0 (curl; Ω) H(div 0 ; Ω) Maxwell s equation Poisson equation + RCCP

22 Reduced Curl-Curl Problem (RCCP) Therefore, we can focus on the reduced curl-curl problem. Advantage: RCCP behaves like an elliptic problem, which greatly simplifies the analysis. Disadvantage: It is difficult to construct finite element subspaces for H 0 (curl; Ω) H(div 0 ; Ω). This difficulty can be overcome by nonconforming methods. S.C.Brenner, F. Li, and L.-Y. Sung. 2007, 2008

23 Reduced Curl-Curl Problem (RCCP) Therefore, we can focus on the reduced curl-curl problem. Advantage: RCCP behaves like an elliptic problem, which greatly simplifies the analysis. Disadvantage: It is difficult to construct finite element subspaces for H 0 (curl; Ω) H(div 0 ; Ω). Observation: The zero divergence condition in the RCCP leads to a larger condition number for the discrete problem (like a 4 th order problem).

24 Curl-Curl and Grad-Div Problem (CCGD) Define H(div; Ω) = {v [L 2 (Ω)] 2 : v L 2 (Ω)}

25 Curl-Curl and Grad-Div Problem (CCGD) Define H(div; Ω) = {v [L 2 (Ω)] 2 : v L 2 (Ω)} Find u H 0 (curl; Ω) H(div; Ω) such that ( u, v) + γ( u, v) + α(u, v) = (f, v) v H 0 (curl; Ω) H(div; Ω) where α R and γ > 0 are constants, f [L 2 (Ω)] 2.

26 Curl-Curl and Grad-Div Problem (CCGD) Define H(div; Ω) = {v [L 2 (Ω)] 2 : v L 2 (Ω)} Find u H 0 (curl; Ω) H(div; Ω) such that ( u, v) + γ( u, v) + α(u, v) = (f, v) v H 0 (curl; Ω) H(div; Ω) When f = 0, the solution u of CCGD, if exists, belongs to the space H(div 0 ; Ω), and it is also the solution of the non-elliptic curl-curl problem.

27 Some History The elliptic curl-curl and grad-div problem was discretized by H 1 conforming nodal finite elements. J.L. Coulomb

28 Some History The elliptic curl-curl and grad-div problem was discretized by H 1 conforming nodal finite elements. J.L. Coulomb The space [H 1 (Ω)] 2 [H 0 (curl; Ω) H(div; Ω)] turns out to be a proper closed subspace of H 0 (curl; Ω) H(div; Ω). Therefore any H 1 conforming finite element method for CCGD problem must fail if the solution u does not belong to [H 1 (Ω)] 2, which happens when Ω is non-convex. M. Birman and M. Solomyak M. Costabel

29 Some History Even worse, the solutions obtained by H 1 conforming methods in such situations converge to the wrong solution (the projection of u in [H 1 (Ω)] 2 [H 0 (curl; Ω) H(div; Ω)]).

30 Some History Even worse, the solutions obtained by H 1 conforming methods in such situations converge to the wrong solution (the projection of u in [H 1 (Ω)] 2 [H 0 (curl; Ω) H(div; Ω)]). Because of this difficulty, the non-elliptic curl-curl problem is usually solved directly using H(curl) conforming finite elements. J.C. Nédélec. 1980, 1986.

31 Our Goal: Solve the elliptic CCGD problem by nonconforming finite element methods. The finite element space V h [H 0 (curl; Ω) H(div; Ω)].

32 Well-Posedness of CCGD When α > 0, f [L 2 (Ω)] 2, the CCGD problem has a unique solution in H 0 (curl; Ω) H(div; Ω). (Riesz Representation Theorem)

33 Well-Posedness of CCGD When α > 0, f [L 2 (Ω)] 2, the CCGD problem has a unique solution in H 0 (curl; Ω) H(div; Ω). (Riesz Representation Theorem) When α 0, the CCGD problem is well-posed as long as α λ γ,j, where 0 λ γ,1 λ γ,2 are the eigenvalues defined by the following eigenproblem: ( w, v) + γ( w, v) = λ γ,j (w, v) (Fredholm theory) v H 0 (curl; Ω) H(div; Ω)

34 Elliptic Regularity of CCGD The regularity of the solution of CCGD problem is closely related to the regularity of the Laplace operator with homogeneous Dirichlet or Neumann boundary conditions. P. Grisvard. 1985, M. Dauge S.A. Nazarov and B.A. Plamenevsky

35 Elliptic Regularity of CCGD For simplicity, we assume Ω is simply connected. For u H 0 (curl; Ω) H(div; Ω), recall Helmholtz decomposition, u = ů + ζ where ů H 0 (curl; Ω) H(div 0 ; Ω) and ζ H 1 0(Ω).

36 Elliptic Regularity of CCGD For simplicity, we assume Ω is simply connected. For u H 0 (curl; Ω) H(div; Ω), recall Helmholtz decomposition, u = ů + ζ where ů H 0 (curl; Ω) H(div 0 ; Ω) and ζ H 1 0(Ω). ů = 0 ů = ψ for some ψ H 1 (Ω) Therefore u = ψ + ζ

37 Elliptic Regularity of CCGD If we apply to both sides of u = ψ + ζ, u = ( ψ) + ( ζ) u = ζ

38 Elliptic Regularity of CCGD If we apply to both sides of u = ψ + ζ, u = ( ψ) + ( ζ) u = ζ So the function ζ H0(Ω) 1 satisfies the following Dirichlet boundary value problem: ζ = u in Ω, ζ = 0 on Ω.

39 Elliptic Regularity of CCGD If we apply to both sides of u = ψ + ζ, u = ( ψ) + ( ζ) u = ψ

40 Elliptic Regularity of CCGD If we apply to both sides of u = ψ + ζ, u = ( ψ) + ( ζ) u = ψ Then ψ H 1 (Ω) satisfies the following Neumann boundary value problem: ψ = u in Ω, ψ n = 0 on Ω.

41 Elliptic Regularity of CCGD u = u R + u S, where u R [H 2 ε (Ω)] 2 is the regular part,

42 Elliptic Regularity of CCGD u = u R + u S, where u R [H 2 ε (Ω)] 2 is the regular part, and u S is supported near the corners c l (1 l L) of Ω.

43 Elliptic Regularity of CCGD u = u R + u S, where u R [H 2 ε (Ω)] 2 is the regular part, and u S is supported near the corners c l (1 l L) of Ω. j N j(π/ω l ) (0,2)\{1} u S = ν l,j r j(π/ω l) 1 l [ sin ( j(π/ω l ) 1 ) ] θ l cos ( j(π/ω l ) 1 ) θ l. where ν l,j are constants, (r l, θ l ) are the polar coordinates at c l such that the two edges at c l are given by θ l = 0, ω l.

44 Elliptic Regularity of CCGD u = u R + u S, where u R [H 2 ε (Ω)] 2 is the regular part, and u S is supported near the corners c l (1 l L) of Ω. j N j(π/ω l ) (0,2)\{1} u S = ν l,j r j(π/ω l) 1 l [ sin ( j(π/ω l ) 1 ) ] θ l cos ( j(π/ω l ) 1 ) θ l. where ν l,j are constants, (r l, θ l ) are the polar coordinates at c l such that the two edges at c l are given by θ l = 0, ω l. Near the corners c l where ω l > π, the singular part 2 u S / H 2 (Ω), therefore the triangulation needs to be graded near such corners.

45 Graded Triangulation T h To recover optimal a priori error estimates in the presence of singularities, graded meshes around the corners c 1,..., c L of Ω are needed.

46 Graded Triangulation T h To recover optimal a priori error estimates in the presence of singularities, graded meshes around the corners c 1,..., c L of Ω are needed. [ L ] h T = diam T h c l c(t ) 1 µ l l=1 c(t ) : the centroid of T = hφ µ (T )

47 Graded Triangulation T h To recover optimal a priori error estimates in the presence of singularities, graded meshes around the corners c 1,..., c L of Ω are needed. [ L ] h T = diam T h c l c(t ) 1 µ l l=1 = hφ µ (T ) The grading parameters µ 1,..., µ L are chosen according to µ l = 1 if ω l π 2 µ l < π 2ω l if ω l > π 2

48 Graded Triangulation T h The graded meshes on an L-shaped domain are depicted in the figures below where the grading parameter equals 1/3 at the re-entrant corner and 1 at all the other corners.

49 Numerical Space Crouzeix-Raviart nonconforming P 1 vector fields (1973) V h = {v [L 2 (Ω)] 2 : v [P 1 (T )] 2, T T h, v is continuous at interior m e, n v = 0 at m e Ω} where m e is the midpoint of e.

50 Jumps of Vector Fields Let e be an interior edge shared by the triangles T 1, T 2. Then we define on e, [[n v]] = n 1 v 1 + n 2 v 2 [[n v]] = n 1 v 1 + n 2 v 2 v i = v Ti, n i = unit normal of e pointing outside T i e T 2 n 1 T 1 n 2

51 Jumps of Vector Fields Let e be an interior edge shared by the triangles T 1, T 2. Then we define on e, [[n v]] = n 1 v 1 + n 2 v 2 [[n v]] = n 1 v 1 + n 2 v 2 v i = v Ti, n i = unit normal of e pointing outside T i On an edge e along Ω, we define [[n v]] = n (v e ) (n = the unit normal of e pointing outside Ω)

52 A Nonconforming Finite Element Method Find u h V h such that a h (u h, v) = (f, v), v V h where a h (w, v) =( h w, h v) + γ( h w, h v) + α(w, v) + [Φ µ (e)] 2 [n w ][n v ]ds e e E e h + [Φ µ (e)] 2 [n w ][n v ]ds e e Eh i e and the edge weight Φ µ (e) = Π L l=1 c l m e 1 µ l.

53 A Nonconforming Finite Element Method Find u h V h such that a h (u h, v) = (f, v), v V h where a h (w, v) =( h w, h v) + γ( h w, h v) + α(w, v) + [Φ µ (e)] 2 [n w ][n v ]ds e e E e h + [Φ µ (e)] 2 [n w ][n v ]ds e e Eh i e The last two terms that control the consistency error are crucial for the convergence of the scheme.

54 A Nonconforming Finite Element Method Errors of the scheme without the consistency terms on the square (0, 0.5) 2, with uniform meshes and γ = 1. h u u h L2 (Ω) u L2 (Ω) u u h curl u curl u u h div u div α = 1 1/ E E E 01 1/ E E E 01 1/ E E E 01 1/ E E E 01 α = 1 1/ E E E 01 1/ E E E 01 1/ E E E 01 1/ E E E 01

55 A Nonconforming Finite Element Method We measure the discretization error in the L 2 norm and the mesh-dependent energy norm h, which is defined by v 2 h = h v 2 L 2 (Ω) + γ h v 2 L 2 (Ω) + v 2 L 2 (Ω) + e E h [Φ µ (e)] 2 e [[n v]] 2 L 2 (Ω) + e E i h [Φ µ (e)] 2 [[n v]] 2 L e 2 (Ω)

56 A Nonconforming Finite Element Method In the case of α > 0, we have the following convergence theorem. Theorem Let α be positive. The following estimates hold for the solution u h of the discrete problem: u u h h C ε h 1 ε f L2 (Ω), u u h L2 (Ω) C ε h 2 ε f L2 (Ω).

57 A Nonconforming Finite Element Method In the case of α 0, we have the following convergence theorem. Theorem Assume that α 0 is not an eigenvalue of CCGD problem. There exists a positive number h such that the discrete problem is uniquely solvable for all h h, and we have the following discretization error estimates: u u h h C ε h 1 ε f L2 (Ω), u u h L2 (Ω) C ε h 2 ε f L2 (Ω). S.C. Brenner, J. Cui, F. Li, and L.-Y. Sung. A nonconforming finite element method for a two-dimensional curl-curl and grad-div problem. Numer. Math., 109, 2008.

58 A Nonconforming Finite Element Method Outline of the proof For α > 0, we first derive an abstract error estimate: u u h h inf v V h u v h + C sup w V h \{0} a h (u u h, w) w h

59 A Nonconforming Finite Element Method Outline of the proof For α > 0, we first derive an abstract error estimate: u u h h inf v V h u v h + C sup w V h \{0} a h (u u h, w) w h inf v V h u v h C ε h 1 ε f L2 (Ω) Interpolation error estimate The properties of T h The elliptic regularity of u

60 A Nonconforming Finite Element Method Outline of the proof For α > 0, we first derive an abstract error estimate: u u h h inf v V h u v h + C sup w V h \{0} a h (u u h, w) w h inf v V h u v h C ε h 1 ε f L2 (Ω) sup w V h \{0} a h (u u h, w) w h C ε h 1 ε f L2 (Ω) The weak continuity of the nonconforming P 1 vector fields. The consistency terms involving the jumps.

61 A Nonconforming Finite Element Method Outline of the proof For α > 0, we first derive an abstract error estimate: u u h h inf v V h u v h + C sup w V h \{0} a h (u u h, w) w h inf v V h u v h C ε h 1 ε f L2 (Ω) Therefore sup w V h \{0} a h (u u h, w) w h C ε h 1 ε f L2 (Ω) u u h h C ε h 1 ε f L2 (Ω)

62 A Nonconforming Finite Element Method We then prove L 2 discretization error estimate: u u h L2 (Ω) C ε h 2 ε f L2 (Ω) A standard duality argument. The elliptic regularity of u.

63 A Nonconforming Finite Element Method We then prove L 2 discretization error estimate: u u h L2 (Ω) C ε h 2 ε f L2 (Ω) A standard duality argument. The elliptic regularity of u. For α 0, we use the approach of Schatz for indefinite problems in the analysis. A. Schatz

64 Numerical Experiments We examine the convergence behavior on the L-shaped domain ( 0.5, 0.5) 2 \ [0, 0.5] 2.

65 Numerical Experiments We examine the convergence behavior on the L-shaped domain ( 0.5, 0.5) 2 \ [0, 0.5] 2. Curl-curl and grad-div problem Find u H 0 (curl; Ω) H(div; Ω) such that ( u, v) + γ( u, v) + α(u, v) = (f, v) where α R and γ = 1. v H 0 (curl; Ω) H(div; Ω)

66 Numerical Experiments We examine the convergence behavior on the L-shaped domain ( 0.5, 0.5) 2 \ [0, 0.5] 2. The exact solution is chosen to be ( ( 2 u = r 2/3 cos 3 θ π ) ) φ(r/0.5), 3 where (r, θ) are the polar coordinates at the origin and the cut-off function is given by 1 r (r 0.75) 3 φ(r) = [ (r 0.75) + 12(r 0.75) 2] 0.25 r r 0.75

67 h u u h L2 (Ω) u L2 (Ω) order u u h h u h order α = 0 1/8 3.24E E / E E / E E / E E α = 1 1/8 2.82E E / E E / E E / E E α = 1 1/8 3.85E E / E E / E E / E E

68 Interior Penalty Method

69 Interior Penalty Method Numerical space Discontinuous P 1 vector fields ˆV h = {v [L 2 (Ω)] 2 : v [P 1 (T )] 2, T T h }.

70 Interior Penalty Method Numerical space Discontinuous P 1 vector fields ˆV h = {v [L 2 (Ω)] 2 : v [P 1 (T )] 2, T T h }. Remark: By removing the weak continuity condition of the vector fields, the interior penalty method can be applied to meshes with hanging nodes.

71 Interior Penalty Method Find u h ˆV h such that â h (u h, v) = (f, v), v ˆV h, where â h (w, v) = ( h w, h v) + γ( h w, h v) + α(w, v) + [Φ µ (e)] 2 [[n w]] [[n v]]ds e e E e h + [Φ µ (e)] 2 [[n w]][[n v]]ds e e Eh i e + h 2 1 (Π 0 e e[[n w]]) (Π 0 e[[n v]])ds e E h + h 2 e E i h 1 e e e (Π 0 e[[n w]])(π 0 e[[n v]])ds,

72 Interior Penalty Method Find u h ˆV h such that â h (u h, v) = (f, v), v ˆV h, where â h (w, v) = ( h w, h v) + γ( h w, h v) + α(w, v) + [Φ µ (e)] 2 [[n w]] [[n v]]ds e e E e h + [Φ µ (e)] 2 [[n w]][[n v]]ds e e Eh i e + h 2 1 (Π 0 e e[[n w]]) (Π 0 e[[n v]])ds e E h + h 2 e E i h 1 e e e (Π 0 e[[n w]])(π 0 e[[n v]])ds, Π 0 e is the orthogonal projection from L 2 (e) to the space of constant functions on e.

73 Interior Penalty Method We measure the discretization error in the L 2 norm and the mesh-dependent energy norm h, which is defined by v 2 h = h v 2 L 2 (Ω) + γ h v 2 L 2 (Ω) + v 2 L 2 (Ω) + e E h [Φ µ (e)] 2 e [[n v]] 2 L 2 (e) + e E i h + h 2( e E h 1 e Π0 e[[n v]] 2 L 2 (e) + e E i h [Φ µ (e)] 2 [[n v]] 2 L e 2 (e) 1 ) e Π0 e[[n v]] 2 L 2 (e).

74 Interior Penalty Method Optimal convergence rates in both the energy norm and the L 2 norm are achieved. (Same results as using nonconforming finite element methods) S.C. Brenner, J. Cui and L.-Y. Sung. An interior penalty method for a two-dimensional curl-curl and grad-div problem. ANZIAM J., 50, 2009.

75 Nonconforming meshes A triangular mesh with hanging nodes

76 Nonconforming meshes A triangular mesh with hanging nodes Let e be an edge of a triangle in the above triangulation. Then e E h if and only if (1) it contains at least one hanging node

77 Nonconforming meshes A triangular mesh with hanging nodes Let e be an edge of a triangle in the above triangulation. Then e E h if and only if (1) it contains at least one hanging node (2) it is the common edge of two triangles

78 Nonconforming meshes A triangular mesh with hanging nodes Let e be an edge of a triangle in the above triangulation. Then e E h if and only if (1) it contains at least one hanging node (2) it is the common edge of two triangles (3) it is a boundary edge

79 Nonconforming meshes e PSfrag replacements T +,1 T +,2 n n + T +,4 T T +,3 We define, on edge e, [[n v]] = (n v T ) 3 e + (n + v T+,j ) ej, j=1 [[n v]] = (n v T ) 3 e + (n + v T+,j ) ej. j=1

80 Numerical Experiments We examine the convergence behavior of the interior penalty method on the square domain (0, 1) 2 with nonconforming meshes.

81 Numerical Experiments We examine the convergence behavior of the interior penalty method on the square domain (0, 1) 2 with nonconforming meshes. Curl-curl and grad-div problem Find u H 0 (curl; Ω) H(div; Ω) such that ( u, v) + γ( u, v) + α(u, v) = (f, v) where α R and γ = 1. v H 0 (curl; Ω) H(div; Ω)

82 Numerical Experiments We examine the convergence behavior of the interior penalty method on the square domain (0, 1) 2 with nonconforming meshes. The exact solution u is given by y(1 y) u = x(1 x)

83 h u u h L2 (Ω) u L2 (Ω) order u u h h u h order α = 1 1/8 8.82E E / E E / E E / E E α = 0 1/8 1.28E E / E E / E E / E E α = 1 1/8 2.36E E / E E / E E / E E

84 We want to investigate multigrid methods for the CCGD problem on general domains, where we must use graded meshes.

85 We want to investigate multigrid methods for the CCGD problem on general domains, where we must use graded meshes. Since there are many similarities between nonconforming finite element methods for Maxwell s equations and discontinuous Galerkin (DG) methods for the Poisson problem on graded meshes, we first investigate multigrid algorithms for DG methods.

86 Multigrid Methods for Discontinuous Galerkin (DG) Methods on Graded Meshes

87 Model Problem Find u H0(Ω) 1 such that u v dx = Ω Ω f v dx v H 1 0(Ω) where Ω R 2 is a bounded polygonal domain with reentrant corners and f L 2,µ (Ω). (Weak form of the Poisson problem with homogeneous Dirichlet boundary condition)

88 Weighted Sobolev Space Let ω 1,..., ω L be the interior angles at the corners c 1,..., c L of the bounded polygonal domain Ω. Let the parameters µ 1,..., µ l be chosen according to µ l = 1 if ω l < π µ l < π ω l if ω l > π

89 Weighted Sobolev Space Let ω 1,..., ω L be the interior angles at the corners c 1,..., c L of the bounded polygonal domain Ω. Let the parameters µ 1,..., µ l be chosen according to µ l = 1 if ω l < π µ l < π ω l if ω l > π The grading strategy is different from that for the CCGD problem.

90 Weighted Sobolev Space Let ω 1,..., ω L be the interior angles at the corners c 1,..., c L of the bounded polygonal domain Ω. Let the parameters µ 1,..., µ l be chosen according to µ l = 1 if ω l < π µ l < π ω l if ω l > π The weight function φ µ is defined by φ µ (x) = L l=1 x c l 1 µ l

91 Weighted Sobolev Space We define L 2,µ (Ω) := {f L 2,loc (Ω) : f L2,µ (Ω) < } where f L2,µ (Ω) = ( Ω φ 2 µ(x)f 2 (x) dx) 1/2

92 Graded Triangulation T h The graded meshes on an L-shaped domain are depicted in the figures below where the grading parameter equals 2/3 at the re-entrant corner and 1 at all the other corners. Numerical Space: V h = {v L 2 (Ω) : v T = v T P 1 (T ) T T h }

93 Set-up for DG Method Let e be an interior edge shared by the triangles T 1, T 2. Then we define on e, [[v]] = v 1 n 1 + v 2 n 2 {{ v}} = 1 ( ) v1 + v 2 2 v i = v Ti, n i = unit normal of e pointing outside of T i e T 2 n 1 T 1 n 2

94 Set-up for DG Method Let e be an interior edge shared by the triangles T 1, T 2. Then we define on e, [[v]] = v 1 n 1 + v 2 n 2 {{ v}} = 1 ( ) v1 + v 2 2 v i = v Ti, n i = unit normal of e pointing outside of T i On an edge e along Ω, we define [[v]] = (v e )n and {{ v}} = ( v) e (n = the unit normal of e pointing outside Ω)

95 Symmetric Interior Penalty (SIP) Method Find u h V h such that a h (u h, v) = fv dx Ω a h (w, v) = T T h T e E h e v V h w v dx e E h {{ v}} [[w]] ds + η e E h 1 e e {{ w}} [[v]] ds e [[w]] [[v]] ds Douglas and Dupont M.F. Wheeler D.N. Arnold

96 Symmetric Interior Penalty (SIP) Method The SIP method is consistent. The SIP method is stable for η η > 0.

97 Symmetric Interior Penalty (SIP) Method The SIP method is consistent. The SIP method is stable for η η > 0. Consequently, we have the abstract error estimate where u u h h C inf v V h u v h, v 2 h = T T h v 2 L 2 (T ) + η 1 e E h e {{ v}} 2 L 2 (e) +2η e E h e 1 [[v]] 2 L 2 (e).

98 Symmetric Interior Penalty (SIP) Method Lemma Let f L 2,µ (Ω) and u H 1 0(Ω) be the solution of model problem. Then u Π h u h Ch f L2,µ (Ω) Π h : C(Ω) V h is the nodal interpolation operator for the conforming P 1 finite element. i.e., Π h u V h H 1 (Ω) agrees with u at the vertices of the triangles of T h. This lemma extends the classical result of Babushka, Kellogg, and Pitkaranta (1979) for conforming finite elements.

99 Symmetric Interior Penalty (SIP) Method Theorem Let f L 2,µ (Ω), u be the solution of the model problem and u h be the solution of the SIP method associated with the graded mesh T h, then u u h L2 (Ω) + h u u h h Ch 2 f L2,µ (Ω) S.C. Brenner, J. Cui, and L.-Y. Sung. Multigrid methods for the symmetric interior penalty method on graded meshes. Numer. Lin. Alg. Appl., 2009.

100 Symmetric Interior Penalty (SIP) Method Theorem Let f L 2,µ (Ω), u be the solution of the model problem and u h be the solution of the SIP method associated with the graded mesh T h, then u u h L2 (Ω) + h u u h h Ch 2 f L2,µ (Ω) We can extend the results to a class of symmetric, consistent and stable DG methods. S.C. Brenner, J. Cui. and T. Gudi, and L.-Y. Sung. Multigrid algorithms for symmetric discontinuous Galerkin methods on graded meshes. submitted, 2009.

101 Numerical Results Energy errors and L 2 errors for L-shaped domain using LDG (η = 1, left) and SIP (η = 10, right) Error 10 3 Error Energy Error L2 Error 10 4 Energy Error L2 Error Mesh Size Mesh Size

102 Numerical Results Energy errors and L 2 errors for L-shaped domain using Bassi et al. (η = 4, left) and Brezzi et al. (η = 1, right) Error 10 3 Error Energy Error L2 Error 10 4 Energy Error L2 Error Mesh Size Mesh Size

103 Multigrid Method

104 Multigrid Method The multigrid method provides an optimal order (O(n)) algorithm for solving elliptic boundary value problems, i.e., the computational cost is proportional to the number of unknowns. W. Hackbush S.F. McCormick J.H. Bramble and X. Zhang S.C. Brenner and L.R. Scott

105 Multigrid Method The multigrid method provides an optimal order (O(n)) algorithm for solving elliptic boundary value problems, i.e., the computational cost is proportional to the number of unknowns. The multigrid method has two main ingredients: Smoothing on the current grid (damping out the oscillatory part of error) Error correction on a coarse grid (using the iteration scheme developed on the coarse grid)

106 Effect of Smoothing Step Original error After smoothing

107 Multigrid for the SIP Method Let V k be the discontinuous P 1 finite element space associated with T k.

108 Multigrid for the SIP Method Let V k be the discontinuous P 1 finite element space associated with T k. The k-th level SIP method is: Find u k V k such that where a k (w, v) = a k (u k, v) = T T k T e E k e Ω fv dx v V k, w v dx e E k {{ v}} [[w]] ds + η e E k 1 e e {{ w}} [[v]] ds e [[w]] [[v]] ds

109 Multigrid for the SIP Method Let A k : V k V k and f k V k be defined by A k w, v = a k (w, v) v, w V k f k, v = f v dx v V k Ω

110 Multigrid for the SIP Method Let A k : V k V k and f k V k be defined by A k w, v = a k (w, v) v, w V k f k, v = f v dx v V k Ω Then the k th level discrete problem is given by A k u k = f k We consider multigrid methods for equations of the form A k z = g

111 Intergrid Transfer Operators intergrid transfer operators smoothing scheme

112 Intergrid Transfer Operators intergrid transfer operators smoothing scheme Coarse-to-fine I k k 1 : V k 1 V k is the natural injection I k k 1v = v v V k 1 Fine-to-coarse I k 1 k : V k V k 1 is the transpose of I k k 1 I k 1 k w, v = w, I k k 1v w V k, v V k 1

113 A Preconditioned Smoothing Scheme The operator B k : V k V k is defined by B k w, v = h 2 k w(m)v(m) v, w V k, m M T T T k where M T is the set of midpoints of the three edges of T.

114 A Preconditioned Smoothing Scheme The operator B k : V k V k is defined by B k w, v = h 2 k w(m)v(m) v, w V k, m M T T T k where M T is the set of midpoints of the three edges of T. We use a preconditioned Richardson relaxation scheme z new = z old + (λh 2 k)b 1 k (g A kz old ) for the equation A k z = g as the smoother.

115 A Preconditioned Smoothing Scheme The operator B k : V k V k is defined by B k w, v = h 2 k w(m)v(m) v, w V k, m M T T T k where M T is the set of midpoints of the three edges of T. We use a preconditioned Richardson relaxation scheme z new = z old + (λh 2 k)b 1 k (g A kz old ) for the equation A k z = g as the smoother. We can choose the (constant) damping factor λ so that ρ(λh 2 kb 1 k A k) < 1 for k 0

116 Multigrid Algorithms The W -cycle algorithm for A k z = g is defined recursively. For k = 0, MG W (k, g, z 0, m 1, m 2 ) = A 1 0 g.

117 Multigrid Algorithms The W -cycle algorithm for A k z = g is defined recursively. For k = 0, MG W (k, g, z 0, m 1, m 2 ) = A 1 0 g. For k 1, we proceed in three steps. Pre-smoothing Apply m 1 preconditioned relaxation steps with initial guess z 0 to obtain z m1.

118 Multigrid Algorithms The W -cycle algorithm for A k z = g is defined recursively. For k = 0, MG W (k, g, z 0, m 1, m 2 ) = A 1 0 g. For k 1, we proceed in three steps. Pre-smoothing Apply m 1 preconditioned relaxation steps with initial guess z 0 to obtain z m1. Coarse-grid correction Apply the coarse grid algorithm twice to the coarse grid residual equation A k 1 e k 1 = I k 1 k (g A k z m1 ) with initial guess 0 to obtain the coarse grid correction q and let z m1 +1 = z m1 + I k k 1 q

119 Multigrid Algorithms Post-smoothing Apply m 2 preconditioned relaxation steps with initial guess z m1 +1 to obtain z m1 +m Final output MG W (k, g, z 0, m 1, m 2 ) = z m1 +m 2 +1

120 Multigrid Algorithms Post-smoothing Apply m 2 preconditioned relaxation steps with initial guess z m1 +1 to obtain z m1 +m Final output MG W (k, g, z 0, m 1, m 2 ) = z m1 +m 2 +1 The V -cycle algorithm for A k z = g is obtained by applying the coarse grid algorithm only once in the coarse grid correction step.

121 Multigrid Algorithms Post-smoothing Apply m 2 preconditioned relaxation steps with initial guess z m1 +1 to obtain z m1 +m Final output MG W (k, g, z 0, m 1, m 2 ) = z m1 +m 2 +1 The V -cycle algorithm for A k z = g is obtained by applying the coarse grid algorithm only once in the coarse grid correction step. The F -cycle algorithm for A k z = g is obtained by applying the coarse grid algorithm once followed by a V -cycle algorithm in the coarse grid correction step.

122 Convergence Analysis The convergence analysis for W -cycle algorithm uses the error estimates of the SIP method on graded meshes. We follow the ideas in the paper by R.E. Bank and T.F. Dupont (1981) and H. Yserentant (1986). Their approaches are modified since we are considering nonconforming DG method.

123 Mesh-Dependent Norms Let the mesh-dependent norms v j,k for j = 0, 1, 2 and k 1 be defined by v j,k = B k (B 1 k A k) j v, v v V k, k 1.

124 Mesh-Dependent Norms Let the mesh-dependent norms v j,k for j = 0, 1, 2 and k 1 be defined by v j,k = B k (B 1 k A k) j v, v v V k, k 1. In particular, we have v 2 1,k = A k v, v = a k (v, v) v V k, v 2 0,k = B kv, v v 2 L 2, µ (Ω) v V k, where v L2, µ (Ω) = ( Ω φ 2 µ (x)v 2 (x)dx) 1/2

125 Convergence of W -Cycle Algorithm Theorem The output MG W (k, g, z 0, m 1, m 2 ) of the W -cycle algorithm applied to A k z = g satisfies the following estimate: z MG W (k, g, z 0, m 1, m 2 ) 1,k C [(1 + m 1 )(1 + m 2 )] 1/2 z z 0 1,k, where the positive constant C is independent of k, provided that m 1 + m 2 is greater than a positive integer m that is also independent of k. S.C. Brenner, J. Cui, and L.-Y. Sung. Multigrid methods for the symmetric interior penalty method on graded meshes. Numer. Lin. Alg. Appl., 2009.

126 Convergence of V, F -Cycle Algorithms The convergence analysis of V -cycle and F -cycle algorithms requires the additive multigrid theory S.C. Brenner. 2002, 2004.

127 Convergence of V, F -Cycle Algorithms The convergence analysis of V -cycle and F -cycle algorithms requires the additive multigrid theory S.C. Brenner. 2002, By using weighted Sobolev spaces and graded meshes, we can treat the convergence of V -cycle and F -cycle algorithms within the framework of problems with full elliptic regularity.

128 Convergence of V, F -Cycle Algorithms Theorem The output MG V (k, g, z 0, m, m) of the V -cycle algorithm applied to A k z = g satisfies the following estimate: z MG V (k, g, z 0, m, m) 1,k C m z z 0 1,k, where the positive constant C is independent of k, provided that the number of smoothing steps m is greater than a positive integer m that is also independent of k.

129 Convergence of V, F -Cycle Algorithms Theorem The output MG V (k, g, z 0, m, m) of the V -cycle algorithm applied to A k z = g satisfies the following estimate: z MG V (k, g, z 0, m, m) 1,k C m z z 0 1,k, where the positive constant C is independent of k, provided that the number of smoothing steps m is greater than a positive integer m that is also independent of k. Similar result is obtained for the F -cycle algorithm.

130 Convergence of V, F -Cycle Algorithms Theorem The output MG V (k, g, z 0, m, m) of the V -cycle algorithm applied to A k z = g satisfies the following estimate: z MG V (k, g, z 0, m, m) 1,k C m z z 0 1,k, where the positive constant C is independent of k, provided that the number of smoothing steps m is greater than a positive integer m that is also independent of k. Similar result is obtained for the F -cycle algorithm. We can extend the results to other DG methods. S.C. Brenner, J. Cui. and T. Gudi, and L.-Y. Sung

131 Numerical Results W -cycle contraction numbers on the L-shaped domain using SIP with λ = 1/10, m 1 = m 2 = m. k = 1 k = 2 k = 3 k = 4 k = 5 k = 6 k = 7 m = m = m = m = m = m = m = m =

132 Numerical Results F -cycle contraction numbers on the L-shaped domain using SIP with λ = 1/10, m 1 = m 2 = m. k = 1 k = 2 k = 3 k = 4 k = 5 k = 6 k = 7 m = m = m = m = m = m = m = m =

133 Numerical Results V -cycle contraction numbers on the L-shaped domain using SIP with λ = 1/10, m 1 = m 2 = m. k = 1 k = 2 k = 3 k = 4 k = 5 k = 6 k = 7 m = m = m = m = m = m = m = m =

134 Multigrid Methods for CCGD Problem

135 Multigrid Methods for CCGD Problem We have investigated W -cycle multigrid algorithm for the nonconforming method using Crouzeix-Raviart finite elements for the CCGD problem on the unit square

136 Multigrid Methods for CCGD Problem We have investigated W -cycle multigrid algorithm for the nonconforming method using Crouzeix-Raviart finite elements for the CCGD problem on the unit square The operator B k = h 2 k Id k is used in the smoothing steps, where Id k is the identity operator on V k.

137 Multigrid Methods for CCGD Problem We have investigated W -cycle multigrid algorithm for the nonconforming method using Crouzeix-Raviart finite elements for the CCGD problem on the unit square We define the coarse-to-fine intergrid transfer operator I k k 1 : V k 1 V k in the multigrid algorithm as (I k k 1v)(m e ) = v 1(m e ) + v 2 (m e ) 2 where v i = v Ti, if e is an interior edge of T k shared by the triangles T 1, T 2 T k 1, or (Ik 1v)(m k e ) = 0, if e is a boundary edge of T k.

138 Multigrid Methods for CCGD Problem W-cycle contraction numbers for CCGD problem on unit square with α = 1, γ = 1, λ = 1/20, m 1 = m 2 = m. k = 1 k = 2 k = 3 k = 4 k = 5 k = 6 k = 7 m = m = m = m = m = m = m = m =

139 Concluding Remarks We are working on the convergence analysis of multigrid methods for the CCGD problem. Graded meshes must be used on non-convex domains The elliptic regularity of the CCGD problem is different from the elliptic regularity of the Poisson problem.

140 Concluding Remarks We are working on the convergence analysis of multigrid methods for the CCGD problem. Graded meshes must be used on non-convex domains The elliptic regularity of the CCGD problem is different from the elliptic regularity of the Poisson problem. We propose a new numerical approach for two-dimensional Maxwell s equations that is based on the Hodge decomposition and study multigrid methods for the P 1 finite element method

141 Hodge Decomposition and Maxwell s Equations Consider the reduced curl-curl problem: Find u H 0 (curl; Ω) H(div 0 ; Ω) such that ( u, v) + α(u, v) = (f, v) v H 0 (curl; Ω) H(div 0 ; Ω) S.C. Brenner, F. Li, and L.-Y. Sung. 2007, 2008.

142 Hodge Decomposition and Maxwell s Equations For every u H 0 (curl; Ω) H(div 0 ; Ω), we have u = φ + m j=1 c j ψ j, where φ H 1 (Ω), (φ, 1) = 0, φ/ n = 0 on Ω, m + 1 equals the number of components of Ω, and ψ 1,..., ψ m are harmonic functions such that ( ψ j, v) = 0 v H 1 0(Ω) ψ j Γ0 = 0 ψ j Γk = δ jk for 1 k m

143 Hodge Decomposition and Maxwell s Equations For every u H 0 (curl; Ω) H(div 0 ; Ω), we have u = φ + m j=1 c j ψ j, Idea c 1,..., c m and φ that appear in the Hodge decomposition of u can be computed by solving standard second order scalar elliptic boundary value problems

144 Hodge Decomposition and Maxwell s Equations Using a P 1 finite element method, we obtain the quasi-optimal error estimates for both u and u in the L 2 norm on graded meshes S.C. Brenner, J. Cui, Z. Nan, and L.-Y. Sung. Hodge decomposition and Maxwell s equations. submitted, 2010.

145 Hodge Decomposition and Maxwell s Equations Using a P 1 finite element method, we obtain the quasi-optimal error estimates for both u and u in the L 2 norm on graded meshes S.C. Brenner, J. Cui, Z. Nan, and L.-Y. Sung. Hodge decomposition and Maxwell s equations. submitted, We can prove the uniform convergence of multigrid algorithms for the resulting discrete problem. J. Cui. Multigrid methods for two-dimensional Maxwell s equations on graded meshes. in preparation, 2010.

146 Concluding Remarks and Future Tasks We have studied nonconforming finite element methods for two-dimensional Maxwell s equations with perfectly conducting boundary condition. The numerical schemes can be generalized to other boundary conditions, such as the impedance boundary condition. P. Monk

147 Concluding Remarks and Future Tasks We have studied nonconforming finite element methods for two-dimensional Maxwell s equations with perfectly conducting boundary condition. The numerical schemes can be generalized to other boundary conditions, such as the impedance boundary condition. P. Monk The numerical schemes for Maxwell s equations can be applied to solve the curl-curl and grad-div problem, which appears in fluid-structure interaction problems. K.J. Bathe, C. Nitikitpaiboon, and X. Wang D. Boffi and L. Gastaldi

148 Thank You