Basic Principles of Weak Galerkin Finite Element Methods for PDEs

Transcription

1 Basic Principles of Weak Galerkin Finite Element Methods for PDEs Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA Polytopal Element Methods in Mathematics and Engineering

2 References in Weak Galerkin (WG) 1 Search weak Galerkin or Junping Wang on arxiv.org 2 Partial List of Contributors: Xiu Ye, University of Arkansas Chunmei Wang, Georgia Institute of Technology Lin Mu, Michigan State University Guowei Wei, Michigan State University Yanqiu Wang, Oklahoma State University Long Chen, University of California, Irvine Shan Zhao, University of Alabama Ran Zhang, Jilin University, China Ruishu Wang, Jilin University, China Qilong Zhai, Jilin University, China

3 Talk Outline 1 Basics of Weak Galerkin Finite Element Methods (WG-FEM) weak gradient stabilization (weak continuity) implementation and error analysis 2 An Abstract Framework 3 WG-FEM for Model PDEs mixed formulation hybridized WG linear elasticity 4 Primal-Dual Weak Galerkin What is it briefly? The Fokker-Planck equation The Cauchy problem for elliptic equations An abstract framework

4 Related Numerical Methods 1 FEM 2 Stabilized FEMs 3 MFD 4 DG, HDG 5 VEM

5 Second Order Elliptic Problems Find u H0 1 (Ω) such that (a u, v) = (f, v), v H 1 0 (Ω). Procedures in the standard Galerkin finite element method: 1 Partition Ω into triangles or tetrahedra. 2 Construct a subspace, denoted by S h H0 1 (Ω), using piecewise polynomials. 3 Seek for a finite element solution u h from S h such that (a u h, v) = (f, v) v S h.

6 An Out-of-Box Thinking Replace u h and v by any distribution, and u h and v by another distribution, say w v as the generalized derivative, and seek for a distribution u h such that Main Issues: (a w u h, w v) = (f, v), v S h. 1 Functions in S h are to be more general (as distributions or generalized functions) a good feature 2 The gradient v is computed weakly or as distributions Questionable and fixable? 3 The numerical approximations are stable and convergent questionable, how to fix? 4 The schemes are easy to implement and broadly applicable Ideal, and can be achieved.

7 Motivation for WG The classical gradient u for u C 1 (K) can be computed as u φ = u φ + u(φ n) K for all φ [C 1 (K)] 2. K The integrals on the right-hand side requires only u 0 = u in the interior of K, plus u b = u (trace) on the boundary K. We symbolically have w u φ = u 0 φ + u b φ n K K K Thus, u can be extended to {u 0, u b } with u being extended to w u. K

8 Generalized Weak Derivatives: Foundation of WG Weak Derivative For any u = {u 0 ; u b } with u 0 L 2 (K) and u b L 2 ( K), the generalized weak derivative of u in the direction ν is the following linear functional on H 1 (K): ν u, φ = u 0 ν φ + (n ν)u b φ. K K for all φ H 1 (K). The generalized weak derivative shall be called weak derivative.

9 Weak Functions Weak Functions A weak function on the region K refers to a generalized function v = {v 0, v b } such that v 0 L 2 (K) and v b L 2 ( K). The first component v 0 represents the value of v in the interior of K, and the second component v b represents v on the boundary of K. v b may or may not be related to the trace of v 0 on K. The space of weak functions: W (K) = {v = {v 0, v b } : v 0 L 2 (K), v b L 2 ( K)}.

10 Weak Gradient For any v W (K), the weak gradient of v is defined as a bounded linear functional w v in H 1 (K) whose action on each q H 1 (K) is given by w v, q K := v 0 qdk + v b q nds, K where n is the outward normal direction on K. The weak gradient is identical with the strong gradient for smooth weak functions (e.g., as restriction of smooth functions). K

11 Discrete Weak Gradients For computational purpose, the weak gradient needs to be approximated, which leads to discrete weak gradients, w,r, given by w,r v qdk = v 0 qdk + v b q nds, K K K for all q V (K, r). Here V (K, r) [P r (K)] 2 is a subspace. P r (K) is the set of polynomials on K with degree r 0. V (K, r) does not enter into the degrees of freedom in discretization.

12 Weak Finite Element Spaces T h : polygonal/polytopal partition of the domain Ω, shape regular construct local discrete elements W k (T ) := {v = {v 0, v b } : v 0 P k (T ), v b P k 1 ( T )}. patch local elements together to get a global space S h := {v = {v 0, v b } : {v 0, v b } T W k (T ), T T h }. Weak FE Space Weak finite element space with homogeneous boundary value: S 0 h := {v = {v 0, v b } S h, v b T Ω = 0, T T h }.

13 Weak Finite Element Functions Element Shape Functions for P 1 (K)/P 0 ( K): φ i = {λ i, 0}, i = 1, 2, 3, φ 3+j = {0, τ j }, j = 1, 2,, N, where N is the number of sides. Figure: WG element

14 Shape Regularity for Polytopal Elements F E n e x e A D A e C B Why Shape Regularity? The shape regularity is needed for (1) trace inequality, (2) inverse inequality, and (3) domain inverse inequality.

15 Weak Galerkin Finite Element Formulation WG-FEM Find u h = {u 0 ; u b } S 0 h such that (a w u h, w v) + s(u h, v) = (f, v 0 ), v = {v 0 ; v b } S 0 h, where 1 w v P k 1 (T ) is the discrete weak gradient computed locally on each element, 2 s(, ) is a stabilizer enforcing a weak continuity, 3 the stabilizer s(, ) measures the discontinuity of the finite element solution.

16 Depiction of the General WG Element P j 1 (e) P j 1 (e) P j (T ) P j 1 (e) The polynomial spaces P j 1 (T ) or P j (T ) can be used for the computation of the weak gradient w.

17 Stabilizer Commonly used stabilizer: s(w, v) = ρ T T h h 1 T Q bw 0 w b, Q b v 0 v b T, where Q b is the L 2 projection onto P k 1 (e), e T. Discrete and computation-friendly stabilizer: s(w, v) = ρ (Q b w 0 w b )(x j ) (Q b v 0 v b )(x j ), T T h x j where {x j } is a set of (nodal) points on T.

18 WG from the Minimization Perspective The original problem can be characterized as ( ) 1 u = arg min (a v, v) (f, v) v H0 1(Ω) 2 Weak Galerkin finite element scheme u h = arg min v S 0 h ( 1 2 (a w v, w v) (f, v 0 ) + 1 s(v, v) 2 ).

19 L 2 Projections and Weak Gradients The following is a commutative diagram: H 1 (T ) [L 2 (T )] d Q h w Q h S h (T ) V (r, T ) 0 or equivalently Implication: w (Q h u) = Q h ( u), u H 1 (T ). The discrete weak gradient is a good approximation of the true gradient operator.

20 Error Estimate With the correct regularity assumptions, one has the following optimal order error estimate Q h u u h Q h u u h 1,h Ch k+1 u k+1. Error estimates in negative norms hold true as well. Thus, the WG-FEM solutions are of superconvergent at certain places.

21 Computational Implementation 1 The stiffness matrix can be assembled as the sum of element stiffness matrices 2 WG preserves physical quantities of importance: mass conservation, energy conservation, etc 3 Suitable for parallel computation; element degree of freedoms can be eliminated in parallel 4 Suitable for multiscale analysis 5 Ideal for problems with discontinuous solutions

22 An Abstract Framework Abstract Problem Find u V such that a(u, v) = f (v), v V. In application to PDE, the space V has certain embedded continuities, such as H 1, H(div), H(curl), H 2, or weighted-version of them. Assume V h : finite dimensional spaces that approximate V a h (, ): bilinear forms on V h V h that approximate a(, ) f h : linear functionals on V h that approximate f. s h (, ): stabilizers that provide necessary smoothness In WG for Poisson equation, the first bilinear form refers to a h (u, v) = (a w u, w v), and the second one s h (, ) is the stabilizer.

23 Abstract WG Abstract WG Find u h V h such that Some Assumptions: a h (u h, v) + s h (u h, v) = f h (v), v V h. Regularity: The solution of the abstract problem lies in a subspace H V The (discrete) norm Vh can be extended to H + V h so that the topology of H is given by the family of semi-norms Vh, h (0, h 0 ) Boundedness and Coercivity: The bilinear form is bounded and coercive in V h. a wh (, ) := a h (, ) + s h (, )

24 Almost Consistency The Abstract WG algorithm is said to be almost consistent if there exists a linear projection/interpolation operator Q h : V V h such that for each u f H of the abstract problem, one has Interpolation approximation: Residual consistency: Almost smoothness: lim u f Q h u f Vh = 0 h 0 a h (Q h u f, v) f h (v) lim sup = 0 h 0 v 0,v V h v Vh s h (Q h u f, v) lim sup = 0. h 0 v 0,v V h v Vh

25 Convergence Convergence Assume that the Abstract WG algorithm is almost consistent. Furthermore, assume that a wh (, ) is bounded and coercive in V h, and the solution to the abstract problem lies in the subspace H. Then, we have lim h 0 u f u h Vh = 0. For the second order elliptic problem, the convergence can be interpreted as ( ) Q h ( u f ) w u h 0 + s(u h, u h ) 1 2 = 0. lim h 0

26 WG: a New Paradigm of Discretization Primal Formulation Find u h such that (a w u h, w v) + s(u h, v) = (f, v), v. Key in WG: discrete weak gradient + stabilization to ensure a weak continuity of u h in H 1. Primal-Mixed Formulation Find u h and q h such that (a 1 q h, v) + ( w u h, v) = 0, v s(u h, w) (q h, w w) = (f, w), w. Key in WG: discrete weak gradient + stabilization to provide a weak continuity for u h in H 1.

27 WG: a New Paradigm of Discretization Dual-Mixed Formulation Find q h and u h such that s(q h, v h ) + (a 1 q h, v) (u, w v) = 0, v ( w q h, w) = (f, w), w. Key in WG: discrete weak divergence + stabilization in velocity to ensure a weak continuity with respect to H(div). The space P j+1 (T ) is used for the computation of w v. Lowest order element: pw constant for flux, pw linear for pressure The finite element partition is of general polytopal type.

28 WG Can Be Hybridized Hybridized Dual-Mixed Formulation Find q h, u h, and λ h such that s(q h, v h ) + (a 1 q h, v) (u, w v) + T λ h, v b n T = 0, ( w q h, w) + T σ, q b n T = (f, w). Key in HWG: variable reduction in the sense that q h and u h can be eliminated locally on each element. P j n P j n [P j ] d P j+1 P j n

29 Linear Elasticity Problems Model Problem: Find a displacement vector field u satisfying σ(u) = f, in Ω, u = û, on Γ. Stress-strain relation for linear, homogeneous, and isotropic materials: σ(u) = 2µε(u) + λ( u)i, (Primal Form) Find u [H 1 (Ω)] d satisfying u = û on Γ and 2(µε(u), ε(v)) + (λ u, v) = (f, v), v [H 1 0 (Ω)] d.

30 Linear Elasticity Problems-Mixed Formulation Introducing a pressure variable p = λ u, the elasticity problem can be reformulated as follows: (Mixed Formulation) Find u [H 1 (Ω)] d and p L 2 (Ω) satisfying u = û on Γ, the compatibility condition Ω λ 1 pdx = Γ û nds, 2(µε(u), ε(v)) + ( v, p) = (f, v), v [H 1 0 (Ω)] d, ( u, q) (λ 1 p, q) = 0, q L 2 0(Ω).

31 Weak Divergence Operator The space of weak vector-valued functions in K V (K) = {v = {v 0, v b } : v 0 [L 2 (K)] d, v b [L 2 ( K)] d }. Weak Divergence The weak divergence of v V (K), w v, is a bounded linear functional on H 1 (K), so that its action on any φ H 1 (K) is given by w v, φ K := (v 0, φ) K + v b n, φ K. Discrete Weak Divergence The discrete weak divergence of v V (K), denoted by w,r,k v, is the unique polynomial, satisfying ( w,r,k v, φ) K = (v 0, φ) K + v b n, φ K, φ P r (K).

32 Weak Finite Element Spaces [P k 1 ] 2 + P RM [P k 1 ] 2 + P RM [P k ] 2 [P k 1 ] 2 + P RM w v [P k 1 (T )] 2 2 w v P k 1 (T ).

33 Weak Galerkin Algorithms for Primal Formulation WG-FEM Primal Find u h = {u 0, u b } V h with u b = Q b û on Γ such that for all v = {v 0, v b } V 0 h, T T h 2(µε w (u h ), ε w (v)) T + (λ w u h, w v) T + s(u h, v) = (f, v 0 ). Q b : L 2 projection onto [P k 1 (e)] d + P RM (e) ε w (u) = 1 2 ( w u + w u T ) Stablizer: s(w, v) = T T h h 1 T Q bw 0 w b, Q b v 0 v b T

34 Weak Galerkin Algorithms for Mixed Formulation WG-FEM in Mixed Form Find u h = {u 0, u b } V h and p h W h satisfying u b = Q b û on Γ, the compatibility condition (λ 1 p h, 1) = Γ û nds, and 2(µε w (u h ), ε w (v)) h + s(u h, v) + ( w v, p h ) h = (f, v 0 ), v V 0 h, ( w u h, q) h λ 1 (p h, q) = 0, q W 0 h. W h = {q : q T P k 1 (T ), T T h } W 0 h = W h L 2 0 (Ω) WG-FEM Primal = WG-FEM Mixed The two weak Galerkin Algorithms are equivalent in the sense that the solutions to the two weak Galerkin Algorithms are identical to each other.

35 Error Estimate in a Discrete H 1 -Norm Error Estimates and Convergence in H 1 Let the exact solution be sufficiently smooth such that (u; p) [H k+1 (Ω)] d H k (Ω). Let (u h ; p h ) V h W h be the weak Galerkin finite element solution. Q h u u h +λ 1 2 Qh p p h + Q h p p h 0 Ch k ( u k+1 + p k ), where C is a generic constant independent of (u; p). Consequently, u u h + λ 1 2 p ph + p p h 0 Ch k ( u k+1 + p k ).

36 Error Estimate in L 2 -Norm Error Estimates and Convergence in L 2 Assume that the exact solution is sufficiently smooth such that (u; p) [H k+1 (Ω)] d H k (Ω). Let (u h ; p h ) V h W h be the weak Galerkin finite element solution. Then, under the regularity assumption, there exists a constant C, such that Q 0 u u 0 Ch k+s( u k+1 + p k ). Moreover, u u 0 Ch k+s( u k+1 + p k ).

37 Numerical Results Ω = (0, 1) 2 ( sin(x) sin(y) the exact solution u = 1 ) Table: WG based on {P 1 (T )/P RM (e)}, λ = 1, µ = /h u 0 Q 0 u order u b Q b u order u h Q h u order

38 Numerical Results Table: WG based on {P 1 (T )/P 1 (e)}, λ = 1, µ = h u h Q 0 u order u b Q b u order u h Q h u order

39 Numerical Results: Locking-Free Experiments Ω = (0, 1) 2 the exact solution ( sin(x) sin(y) u = cos(x) cos(y) ) + λ 1 ( x y ) Table: WG based on {P 1 (T )/P RM (e)}, µ = 0.5, and λ = 1. 1 h u h Q 0 u order u b Q b u order u h Q h u order

40 Numerical Results: Locking-Free Experiments Table: WG based on {P 1 (T )/P RM (e)}, µ = 0.5, and λ = 1, 000, h u h Q 0 u order u b Q b u order u h Q h u order

41 Numerical Results: Locking-Free Experiments Table: WG based on {P 1 (T )/P 1 (e)}, µ = 0.5, and λ = 1. 1 h u h Q 0 u order u b Q b u order u h Q h u order

42 Numerical Results: Locking-Free Experiments Table: WG based on {P 1 (T )/P 1 (e)}, µ = 0.5, and λ = 1, 000, h u h Q 0 u order u b Q b u order u h Q h u order

43 The Fokker-Planck Equation describe the time evolution of the probability density function of the velocity of a particle under the influence of drag forces and random forces, as in Brownian motion. assume the stochastic differential equation: dx t = µ(x t, t)dt + σ(x t, t)dw t the probability density f (x, t) for the random vector X t satisfies the Fokker-Planck equation f t + (µf ) 1 2 N ij[d 2 ij f ] = 0, i,j=1 where µ = (µ 1,, µ N ) is the drift vector and D ij (x, t) = is the diffusion tensor. M σ ik (x, t)σ jk (x, t) k=1

44 Model Problem Find u = u(x) satisfying d ij(a 2 ij u) =g, in Ω, i,j=1 u =0, on Ω. assume that a(x) is non-smooth, weak formulation is given by seeking u such that d (u, a ij ijw) 2 = (g, w), i,j=1 v H 2 (Ω) H 1 0 (Ω).

45 A Cauchy Problem for Elliptic Equations The model problem seeks u such that u = f, in Ω, u = 0, on Γ Ω, u n = ψ, on Γ Ω. This is usually an ill-posed problem which does not have a solution or has many solutions. Let Γ c = Ω/Γ. A variational form for this problem seeks u H0,Γ 1 (Ω) such that for all w H 1 0,Γ c (Ω). ( u, w) = (f, w) + ψ, w Γ,

46 An Abstract Problem Let V and W be two Hilbert spaces b(, ) is a bilinear form on V W The inf-sup condition of Babuska and Brezzi is satisfied. The spaces U and V have certain embedded continuities, such as L 2, H 1, H(div), H(curl), H 2, or weighted-version of them. Abstract Problem Find u V such that b(u, w) = f (w) for all w W. Here f is a bounded linear functional on W. Goal: Design finite element methods by using weak Galerkin approach.

47 Specific Examples (Revisited) For the Fokker-Planck, we have V = L 2 and W = H 2 H0 1 and d b(v, w) := (v, a ij ijw). 2 i,j=1 For the Cauchy problem for Poisson equation, V W = H 1 0,Γ (Ω) H1 0,Γ c (Ω) and b(v, w) := ( v, w). Note that the inf-sup condition may not be satisfied.

48 An Abstract Primal-Dual Formulation Primal equation in color blue, Dual equation in color red, They are connected by stabilizers with weak continuity. WG-FEM Find u h V h and λ h W h such that s 1 (u h, v) b h (v, λ h ) = 0, v V h s 2 (λ h, w) + b h (u h, w) = f h (w), w W h. s 1 (, ): s 2 (, ): stabilizer/smoother in V h stabilizer/smoother in W h

49 The Primal-Dual WG for Elliptic Cauchy Problem the b h (, )-form is given by Both stabilizers are given by: b h (v, w) := ( w v, w w), s(u, v) = T T h h 1 T u 0 u b, v 0 v b T +h T n u 0 u gn, n v 0 v gn T. Error estimates and numerical experiments are on the way.

50 Numerical Tests the right-hand side f is given by 2a 11 +2a 22 10a 12 10a sin(30(x 0.5) 2 +30(y 0.5) 2 ) the boundary condition u = x 2 + 2y 2 5xy on Ω Ω = (0, 1) 2 Figure: WG finite element solution with coefficients a 11 = 3, a 12 = a 21 = 1 and a 22 = 2 in mesh size 1.250e-01 (ρ h is piecewise linear function).

51 Numerical Tests Figure: WG finite element solution with coefficients a 11 = 10, a 12 = a 21 = (0.25x) 2 (0.25y) 2, a 22 = 10 in mesh size 1.250e-01 (ρ h is piecewise linear function).

52 Numerical Tests Figure: WG finite element solution with coefficients a 11 = 3, a 12 = a 21 = 1 and a 22 = 2 in mesh size 1.250e-01 (ρ h is a piecewise constant).

53 Numerical Tests Figure: WG finite element solution with coefficients a 11 = 10, a 12 = a 21 = (0.25x) 2 (0.25y) 2, a 22 = 10 in mesh size 1.250e-01 (ρ h is a piecewise constant).

54 Current and Future Research The current and future research projects include 1 WG on polytopal partitions with curved sides, 2 Fokker-Planck equation, 3 Nonlinear PDEs such as MHD and Cahn-Hillard equations, 4 Variational problems where the trial and test spaces are different, but an inf-sup condition is satisfied, 5 Applications and efficient implementation issues.

55 Thanks for your attention!