The Discontinuous Galerkin Finite Element Method

Transcription

1 The Discontinuous Galerkin Finite Element Method Michael A. Saum Department of Mathematics University of Tennessee, Knoxville The Discontinuous Galerkin Finite Element Method p.1/41

2 Overview What is DG? DG Formulation Data Structures Solvers Adaptivity The Discontinuous Galerkin Finite Element Method p.2/41

3 What is DG? The Discontinuous Galerkin (DG) Finite Element Method (FEM) is a variant of the Standard (Continuous) Galerkin (SG) FEM. SG-FEM requires continuity of the solution along element interfaces (edges). DG-FEM does not require continuity of the solution along edges. DG methods have more degrees of freedom (unknowns) to solve for than SG methods. The Discontinuous Galerkin Finite Element Method p.3/41

4 DG Advantages DG methods have a number of advantages over SG methods: Assembly of stiffness matrix is easier to implement. Refinement of triangles is easier to implement. Adaptive methods are more flexible. Natural Hierarchy allows for multilevel methods to be integrated into solvers. DG methods can support high order local approximations that can vary nonuniformly over the mesh. DG methods are readily parallelizable. The Discontinuous Galerkin Finite Element Method p.4/41

5 Model Problem Let Ω R d, d = 2, 3 be a bounded open polyhedral domain with Lipshitz continuous boundary. u = f in Ω u = g D on Γ D (MP) u n = g N on Γ N where Ω := Γ = Γ D Γ N and n is the unit normal vector exterior to Ω. We also assume that µ d 1 (Γ D ) > 0, f L 2 (Ω), g N L 2 (Γ N ). The Discontinuous Galerkin Finite Element Method p.5/41

6 Notation Let Th = {K i : i = 1, 2,..., m h } be a family of star-like partitions of Ω parameterized by 0 < h 1. The elements of Th satisfy the minimal angle condition. Th is locally quasi-uniform. E I = {e = K j K l : µ d 1 ( K j K l ) > 0} E B = {e = K j Ω : µ d 1 ( K j Ω) > 0} e E B, either e Γ D or e Γ N and E = E I E B, where E B = ED B EB N and ED B EB N =. If e E I, then e = K + K for K +, K T h. If e E B, then e = K + Ω K Ω. n + is the unit normal to e that points outward from K +. On Th, for r 2, define the energy space E h and finite element space Vh r by E h = K T h H 2 (K), V r h = K T h P k (K) where P k (K) denotes the space of polynomials of total degree r 1 k 1. The Discontinuous Galerkin Finite Element Method p.6/41

7 DG Formulation First obtain weak formulation by multiplying (MP) by v V r h and integrating over Ω: ( u)v dx = Ω Ω fv dx Now decompose integrals into element contributions and integrate by parts: K T h K K T h u v dx K T h K ( u)v dx = K T h K u n v ds = K T h K K fv dx fv dx The Discontinuous Galerkin Finite Element Method p.7/41

8 DG Formulation, contd. Splitting Edge integrals: K T h Resulting in: u n, v K = e Γ D u n, v + e E I ( u + n +, v e + u n, v e Γ N e ) u + n, v e e K T h ( u, v) K u n, v Γ D e E I u + n +, v e u n +, v e = K T h (f, v) K + g N, v ΓN The Discontinuous Galerkin Finite Element Method p.8/41

9 DG Formulation, contd. Two different ways of working with above internal edge integrals: 1 D. Arnold: ac bd = 2 (a + b)(c d) + 1 (a b)(c + d). 2 G. Baker: ac bd = a(c d) + (a b)d. Define B(u, v) := ( u, v) K K Th F (v) := K Th (f, v) K + g N, v ΓN J(u, v) := where u n u n e u n, v e = 1 2 = u+ n e Γ D + u + n e E I + u n (Baker), and u n e, [v] e (Arnold) and, [v] e = v + v. e The Discontinuous Galerkin Finite Element Method p.9/41

10 SIPG Formulation Leads to the DG formulation of (MP): Find u H 1 E h such that B(u, v) J(u, v) = F (v) v E h Symmetric Interior Penalty Formulation (SIPG) involves modifications: Symmetrization: B(u, v) J(u, v) J(v, u) = F (v) v n, g D Γ D The Discontinuous Galerkin Finite Element Method p.10/41

11 SIPG Formulation, contd. Penalization of jump terms: Let σ > 0 be a penalization parameter Let J σ (u, v) := σ[u], [v] e + σu, v ΓD e E I SIPG Formulation: Find u H 1 E h such that B(u, v) J(u, v) J(v, u) + J σ (u, v) v = F (v) n, g D + σg D, v ΓD Γ D v E h The Discontinuous Galerkin Finite Element Method p.11/41

12 DG FEM Formulation Find u γ h V r h such that a γ h (uγ h, v) = F γ h (v), v V r h where a γ h (uγ h, v) = K T h ( u γ h, v) K e E I E B D ( { n u γ h }, [v] e + { nv}, [u γ h ] e γh 1 e [u γ h ], [v] e ) and F γ h (v) = (f, v) K g D, n v γh 1 e v + g Γ N, v D ΓN K T h The Discontinuous Galerkin Finite Element Method p.12/41

13 Energy Norm The bilinear form a γ h (, ) induces the following norm on E h : v 1,h = ( K T h v 2 0,K + ( ) ) 1/2 h 1 e [v] 2 0,e + h e { n v} 2 0,e e E I E B D Note that a γ h (, ) is symmetric, coercive for σ > σ 0 > 0 for σ 0 large enough. Note that σ = σ(γ, r, h). Common to take σ = γ(r 1) 2 h 1 e, and use the condition γ > γ 0 for γ 0 large enough. The Discontinuous Galerkin Finite Element Method p.13/41

14 Stiffness Matrix Assembly The stiffness matrix has a very nice sparse block structure, consisting of two types of matrix subblocks Diagonal Blocks, which describe interaction of an elements degrees of freedom with itself. Off Diagonal Blocks, which describe interactions of K + dof with K dof through edge e. The triangulation T h has imposed on it the constraint that any element K can at most have 2 neighboring elements K 1, K 2 along edge e. This is the case where one has a hanging node on an edge, it is also called a 1-irregular mesh, or the two-neighbor condition. This results in a maximum block bandwidth of 6 for the stiffness matrix. The Discontinuous Galerkin Finite Element Method p.14/41

15 Data Structures Data objects include TRIANGLE, EDGE, and NODE. Objects stored in one long array of objects for each type via doubly linked list structures. Pointers are used to identify relations between objects. Hierarchial relations are stored in a 4-ary tree structure. PDE data (vectors, stiffness matrix blocks) are stored separately from geometric data. The Discontinuous Galerkin Finite Element Method p.15/41

16 Data Structure Relations IE + BE BLK object layout IE_BLK BE_BLK Lvl 0 Lvl 1 Lvl 2... Lvl l... Lvl L Avail EDGE(0,1,2) (K+,K-) (K+) TRI ENDP(0,1) TRI_BLK offset Used Avail ND(0,1,2) offset offset ND_BLK KTree DIAG_BLK VECTORS OFF_DIAG_BLK Leaf NLeaf Avail ND Hierarchial Tree PDE Data The Discontinuous Galerkin Finite Element Method p.16/41

17 Test Problems - f3 x 2 (1, 1) 1 Γ D Γ D Ω Γ D 0 Γ D 1 { u = 2π 2 sin(πx) sin(πy) u = 0 Exact solution: u = sin(πx) sin(πy). x 1 in Ω on Γ D The Discontinuous Galerkin Finite Element Method p.17/41

18 Test Problems - f4 x 2 (1, 1) 1 Γ D Γ D Ω Γ D x 1 0 Γ D 1 { u = 128π 2 sin(8πx) sin(8πy) u = 0 Exact solution: u = sin(8πx) sin(8πy). in Ω on Γ D The Discontinuous Galerkin Finite Element Method p.18/41

19 Test Problems - f6 Γ D (0.5, 0.5) Γ D (0, 0) Ω Γ D Γ D Γ D { u = 0 in Ω u = r 2/3 sin(2/3θ) on Γ D Exact solution: u = r 2/3 sin(2/3θ). The Discontinuous Galerkin Finite Element Method p.19/41

20 FEM Error A quick way of determining if a FEM is working properly is if one obtains expected reductions in error as one uniformly refines a mesh. For h h/2 uniformly in a mesh with elements of degree p, one expects that ( ) p+1 1 u u h/2 L2 (Ω) u u h L2(Ω) 2 ( ) p 1 u u h/2 H1 (Ω) u u h H1(Ω) 2 As one can see in the following graphs, this is indeed the case for the smooth functions (f3, f4), but not necssarily the case for the point singularity problem (f6). The Discontinuous Galerkin Finite Element Method p.20/41

21 Uniform Refinement Error - f3 f3 e f3 grd e d1 d2 d3 d4 1e 11 1e 08 1e 05 1e 02 1e 09 1e 07 1e 05 1e 03 1e d1 d2 d3 d Triangles Triangles f3 e _{1_h} d1 d2 d3 d4 1e 08 1e 06 1e 04 1e 02 1e Triangles Tue Apr 18 08:08: The Discontinuous Galerkin Finite Element Method p.21/41

22 Uniform Refinement Error - f4 f4 e f4 grd e d1 d2 d3 d4 1e 09 1e 07 1e 05 1e 03 1e 01 1e 05 1e 03 1e 01 1e d1 d2 d3 d Triangles Triangles f4 e _{1_h} d1 d2 d3 d4 1e 05 1e 03 1e 01 1e Triangles Tue Apr 18 08:08: The Discontinuous Galerkin Finite Element Method p.22/41

23 Uniform Refinement Error - f6 f6 e f6 grd e d1 d2 d3 d4 d1 d2 d3 d4 1e 06 1e 05 1e 04 1e 03 1e Triangles Triangles f6 e _{1_h} d1 d2 d3 d Triangles Tue Apr 18 08:08: The Discontinuous Galerkin Finite Element Method p.23/41

24 Linear Solvers Since SIPG produces a symmetric, positive definite linear system to solve, CG and PCG can be used. Due to the natural level based tree hierarchy produced, multigrid can also be used. PCG is used with MG as preconditioner. The previous solution obtained is embedded into the new triangulation to obtain the initial solution for each solve. Point Gauss-Seidel is used as the MG smoother. Local smoothing is implemented to improve solve time, i.e., on a particular level l only dof s associated with levels up to l n are smoothed. Capability exists to implement either V or W cycles. The Discontinuous Galerkin Finite Element Method p.24/41

25 Solver Performance - f3d1 f3d1 Solve Time (s) f3d1 Solver Iterations 1e 03 1e 02 1e 01 1e+00 1e+01 cg mg pcg cg mg pcg 5e+01 5e+02 5e+03 5e+04 5e+01 5e+02 5e+03 5e+04 dof Tue Apr 18 07:43: dof The Discontinuous Galerkin Finite Element Method p.25/41

26 Solver Performance - f3d2 f3d2 Solve Time (s) f3d2 Solver Iterations 1e 03 1e 01 1e+01 cg mg pcg cg mg pcg 1e+02 5e+02 5e+03 5e+04 5e+05 1e+02 5e+02 5e+03 5e+04 5e+05 dof Tue Apr 18 07:43: dof The Discontinuous Galerkin Finite Element Method p.26/41

27 Solver Performance - f3d3 f3d3 Solve Time (s) f3d3 Solver Iterations 5e 03 5e 02 5e 01 5e+00 cg mg pcg cg mg pcg 2e+02 1e+03 5e+03 2e+04 1e+05 5e+05 2e+02 1e+03 5e+03 2e+04 1e+05 5e+05 dof Tue Apr 18 07:43: dof The Discontinuous Galerkin Finite Element Method p.27/41

28 Solver Performance - f3d4 f3d4 Solve Time (s) f3d4 Solver Iterations 5e 03 5e 02 5e 01 5e+00 cg mg pcg cg mg pcg 5e+02 5e+03 5e+04 5e+05 5e+02 5e+03 5e+04 5e+05 dof Tue Apr 18 07:43: dof The Discontinuous Galerkin Finite Element Method p.28/41

29 Adaptivity Uniform refinement is overkill for some problems. The idea of adaptive methods is to utilize some sort of estimator to selectively choose specific elements to refine. Residual based estimators utilize the previously obtained solution to identify candidates for refinement and coarsening. Local Problem based estimators solve local problems usually consisting of each element and its immediate neighbors to identify candidates for refinement and coarsening. An Adaptive Iterations consist of Solve-Estimate-Mark-Refine-Coarsen sequence. Adaptive iterations terminate when the desired tolerance is achieved. The Discontinuous Galerkin Finite Element Method p.29/41

30 Element Refinement DG allows a triangle to undergo regular refinement, i.e., each triangle is divided into four new triangles, each similar to its parent. We impose at most one hanging node per edge. SG doesn t allow hanging nodes to be present. DG refinement allows one to maintain area and normal orientation for the inital mesh triangles only; these quantities can be scaled appropiately for higher level (smaller) elements. Coarsening only occurs when all four children of a triangle are marked for coarsening. The Discontinuous Galerkin Finite Element Method p.30/41

31 A Posteriori Error Estimation The following theorems stated without proof (see Karakashian and Pascal,2004) provide information on residual based a posteriori estimators used to aid in the determination of whether to refine or coarsen individual elements. Theorem. Let e = u u γ h. Then ( e 2 K c h 2 K f + uγ h 2 K K T h K T h + e E I h e [ n u γ h ] 2 e + e E B N h e g N n u γ h 2 e + γ 2 h 1 e [u γ h ] 2 e + γ 2 e E I e ED B ) h 1 e g D u γ h 2 e The Discontinuous Galerkin Finite Element Method p.31/41

32 A Posteriori Error Est., contd Theorem. Suppose f is a piecewise polynomial on T h. Then K T h h 2 K f + u γ h 2 K c e 2 K e = K + K E I h e [ n u γ h ] 2 e c ( e 2 K + + e 2 K ) e = K + Ω E B N for γ large enough γ 2 h 1 e [u γ h ] 2 e +γ 2 e E I e ED B h e g N n u γ h 2 e c e 2 K + h 1 e [g D u γ h ] 2 e c e 2 K K T h The Discontinuous Galerkin Finite Element Method p.32/41

33 f3 Adaptive Meshes The Discontinuous Galerkin Finite Element Method p.33/41

34 f4 Adaptive Meshes The Discontinuous Galerkin Finite Element Method p.34/41

35 f6 Adaptive Meshes The Discontinuous Galerkin Finite Element Method p.35/41

36 f6 Adaptive Meshes - Zoom The Discontinuous Galerkin Finite Element Method p.36/41

37 f3 Adaptive Solution The Discontinuous Galerkin Finite Element Method p.37/41

38 f4 Adaptive Solution The Discontinuous Galerkin Finite Element Method p.38/41

39 f6 Adaptive Solution The Discontinuous Galerkin Finite Element Method p.39/41

40 The Biharmonic Model Problem Let Ω R d, d = 2, 3 be a bounded open polyhedral domain with Lipshitz continuous boundary. 2 u = f u = g D u n = g N in Ω on Γ on Γ where Ω := Γ and n is the unit normal vector exterior to Ω. We also assume that µ d 1 (Γ) > 0, f L 2 (Ω), g N L 2 (Γ). We have created an SIPG implementation for this problem (a different bilinear form). The following solution was obtained using uniform refinement with r = 5 for g D = g N = 0 with exact solution: u = (1 cos(2πx))(1 cos(2πy)) The Discontinuous Galerkin Finite Element Method p.40/41

41 Biharmonic Computed Solution The Discontinuous Galerkin Finite Element Method p.41/41