Weak Galerkin Finite Element Methods and Applications

Weak Galerkin Finite Element Methods and Applications Lin Mu mul1@ornl.gov Computational and Applied Mathematics Computationa Science and Mathematics Division Oak Ridge National Laboratory Georgia Institute of Technology Oct 26-28, 2015 Lin Mu WGFEM and Applications Oct 26-29, 2015 1 / 34

Outline Weak Galerkin Finite Element Methods Motivation Implementation Applications Brinkman Problems Multiscale Weak Galerkin Finite Element Methods Solution Boundedness Summary and Future Work Joint Work with: Dr. Junping Wang, Dr. Guowei Wei, Dr. Xiu Ye Lin Mu WGFEM and Applications Oct 26-29, 2015 2 / 34

WGFEM Table of Contents 1 Weak Galerkin Finite Element Methods Motivation Implementation 2 Applications Brinkman Problems Multiscale Weak Galerkin Finite Element Methods Solution Boundedness 3 Summary and Future Work Summary Future Work Lin Mu WGFEM and Applications Oct 26-29, 2015 3 / 34

WGFEM Motivation Limitations of Continuous Finite Element Methods Lowest order of element does not work P 1 P 0 Continuous element not stable for Stokes problem Difficult to construct high order continuous elements. C 1 element: Argyris element, polynomial with degree 5 Not compatible to modern techniques in scientific computing. hp adaptive technique Hybrid mesh Figure: Argyris Finite Element Figure: Hp adaptive Finite Element Figure: Hybrid mesh Lin Mu WGFEM and Applications Oct 26-29, 2015 4 / 34

WGFEM Implementation Weak Galerkin Formulation The weak form of the PDE: seeking u H0 1 (Ω) satisfying: (a u, v) = (f, v), v H 1 0 (Ω). Weak Galerkin finite element method: seeking u h V h satisfying: (a w u h, w v) + s(u h, v) = (f, v), v V h, where s(u h, v) = K h 1 K u 0 u b, v 0 v b K. 1 0.8 0.6 0.4 0.2 Honeycomb mesh 0 0 0.2 0.4 0.6 0.8 1 Mesh with hanging nodes Lin Mu WGFEM and Applications Oct 26-29, 2015 5 / 34

WGFEM Implementation Weak Galerkin Formulation The weak form of the PDE: seeking u H0 1 (Ω) satisfying: (a u, v) = (f, v), v H 1 0 (Ω). Weak Galerkin finite element method: seeking u h V h satisfying: (a w u h, w v) + s(u h, v) = (f, v), v V h, where s(u h, v) = K h 1 K u 0 u b, v 0 v b K. Like traditional FEM (CG) to compute ( u h, v), for WG we need to compute: 1 (a w u h, w v) = (a w u h, w v) K. K 1 L. Mu, J. Wang, and X. Ye. Weak Galerkin Finite Element Methods on Polytopal Meshes. International Journal of Numerical Analysis and Modeling. 12 (2015), 31-53. Lin Mu WGFEM and Applications Oct 26-29, 2015 6 / 34

WGFEM Implementation Example of WG Element: P 1 (K 0 ), P 0 (e) Calculating: w v [P 0 (K)] 2, with v V h = {(v 0, v b ) : v 0 K P 1 (K 0 ), v b P 0 (e), v b = 0 on Ω} V h = span{φ 0,i, Φ b,j }, i = 1,..., n p, j = 1,..., n e Φ 0,i = Φ b,j = { P 1 (K) on K 0 0, otherwise { 1, on e j 0, otherwise. V 0 e 2 e 1 e 3 e 3 Figure: (a). Φ 0,i is defined on K 0 ; (b). Φ b,j is defined on K. e l e 6 e 5 V b Lin Mu WGFEM and Applications Oct 26-29, 2015 7 / 34

WGFEM Implementation Example of WG Element: P 1 (K 0 ), P 0 (e) Calculating: w v [P 0 (K)] 2, with v V h = {(v 0, v b ) : v 0 K P 1 (K 0 ), v b P 0 (e), v b = 0 on Ω} V h = span{φ 0,i, Φ b,j }, i = 1,..., n p, j = 1,..., n e Φ 0,i = Φ b,j = { P 1 (K) on K 0 0, otherwise { 1, on e j 0, otherwise. V 0 e 2 e 1 e 3 e 3 Figure: (a). Φ 0,i is defined on K 0 ; (b). Φ b,j is defined on K. e l e 6 e 5 V b ( w v, q) K = (v 0, q) K + v b, q n K Lin Mu WGFEM and Applications Oct 26-29, 2015 7 / 34

WGFEM Implementation ( w v, q) K = (v 0, q) K + v b, q n K w Φ 0,i K = 0, w Φ b,j K = e j K n j,k. Lin Mu WGFEM and Applications Oct 26-29, 2015 7 / 34 Example of WG Element: P 1 (K 0 ), P 0 (e) Calculating: w v [P 0 (K)] 2, with v V h = {(v 0, v b ) : v 0 K P 1 (K 0 ), v b P 0 (e), v b = 0 on Ω} V h = span{φ 0,i, Φ b,j }, i = 1,..., n p, j = 1,..., n e Φ 0,i = Φ b,j = { P 1 (K) on K 0 0, otherwise { 1, on e j 0, otherwise. V 0 e 2 e 1 e 3 e 3 Figure: (a). Φ 0,i is defined on K 0 ; (b). Φ b,j is defined on K. e l e 6 e 5 V b

WGFEM Implementation Weak Galerkin Numerical Schemes Weak Galerkin finite element method for Stokes equations: 2 seeking (u h, p h ) V h W h such that for all (v, q) V h W h ( w u h, w v) + s(u h, v) ( w v, p h ) = (f, v) ( w u h, q) = 0. Weak Galerkin finite element method for Biharmonic equations: 3 seeking u h V h satisfying ( w u h, w v) + s(u h, v) = (f, v), v V h. 2 J. Wang, and X. Ye. A Weak Galerkin Finite Element Method for the Stokes Equations, arxiv: 1302.2707. 3 L. Mu, J. Wang, and X. Ye. Weak Galerkin Finite Element Methods for the Biharmonic Equation on Polytopal Meshes. Numerical Methods for Partial Differential Equations, 30 (2014): 1003-1029. Lin Mu WGFEM and Applications Oct 26-29, 2015 8 / 34

Table of Contents 1 Weak Galerkin Finite Element Methods Motivation Implementation 2 Applications Brinkman Problems Multiscale Weak Galerkin Finite Element Methods Solution Boundedness 3 Summary and Future Work Summary Future Work Lin Mu WGFEM and Applications Oct 26-29, 2015 9 / 34

Brinkman Application 1: Brinkman Problems Brinkman equations: ɛ 2 u + u + p = f, in Ω, u = g, in Ω where 0 ɛ 2 1. Stokes equations: u + p = f, in Ω, u = 0, in Ω Darcy equations: u + p = 0, in Ω, u = g, in Ω Lin Mu WGFEM and Applications Oct 26-29, 2015 10 / 34

Brinkman Challenge of algorithm design for the Brinkman equations The main challenge for solving Brinkman equations is in the construction of numerical schemes that are stable for both the Darcy flow and the Stokes flow. Stokes stable elements such as Crouzeix-Raviart element, MINI element and Taylor-Hood element do not work well for the Darcy flow (small ɛ). Darcy stable elements such as RT elements and BDM elements do not work well for the Stokes flow (large ɛ). Lin Mu WGFEM and Applications Oct 26-29, 2015 11 / 34

Brinkman WG methods for the Brinkman equations The weak form of the Brinkman equations: find (u, p) H 1 0 (Ω)d L 2 0 (Ω) s.t. for (v, q) H1 0 (Ω)d L 2 0 (Ω) ɛ 2 ( u, v) + (u, v) ( v, p) = (f, v), ( u, q) = (g, q). Weak Galerkin finite element methods 4 for the Brinkman equations: find (u h, p h ) V h W h s.t. for (v, q) V h W h ɛ 2 ( w u h, w v) + (u 0, v 0 ) + s(u h, v) ( w v, p h ) = (f, v), ( w u h, q) = (g, q), where s(v, w) = K T h h 1 K v 0 v b, w 0 w b K. 4 Mu, L., Wang, J., Ye, X.. A stable numerical algorithm for the Brinkman equations by weak Galerkin finite element methods. Journal of Computational Physics. (2014) Lin Mu WGFEM and Applications Oct 26-29, 2015 12 / 34

Brinkman Uniform convergence rate of WG method Theorem (Mu, Wang, Ye, 2014, Uniform convergence rate) Let (u; p) (H 1 0 (Ω) Hk+1 (Ω)) d L 2 0 (Ω) Hk (Ω) with k 1 be the solution of the Brinkman equations and (u h ; p h ) V h W h be the solutions of the weak Galerkin methods. Then Q h u u h + Q h p p h Ch k ( u k+1 + p k ), Q 0 u u 0 Ch k+1 ( u k+1 + p k ). where C is a constant independent of h and ɛ. Lin Mu WGFEM and Applications Oct 26-29, 2015 13 / 34

Brinkman WG Methods for Brinkman Problems Example 1. Let Ω = (0, 1) 2, u = curl(sin 2 (πx) sin 2 (πy)), and p = sin(πx). Table: Numerical Convergence Test for Crouzeix Raviart element, Raviart Thomas element, and WG element. ɛ 1 2 2 2 4 2 8 0 Crouzeix-Raviart rate H 1,velocity 0.98 0.97 0.74 0.03-0.03 rate L 2,pressure 1.00 0.93 0.98 0.12-0.03 Lin Mu WGFEM and Applications Oct 26-29, 2015 14 / 34

Brinkman WG Methods for Brinkman Problems Example 1. Let Ω = (0, 1) 2, u = curl(sin 2 (πx) sin 2 (πy)), and p = sin(πx). Table: Numerical Convergence Test for Crouzeix Raviart element, Raviart Thomas element, and WG element. ɛ 1 2 2 2 4 2 8 0 Crouzeix-Raviart rate H 1,velocity 0.98 0.97 0.74 0.03-0.03 rate L 2,pressure 1.00 0.93 0.98 0.12-0.03 Raviart-Thomas rate H 1,velocity -0.07-0.07 0.28 0.97 0.97 rate L 2,pressure -0.04 0.08 0.86 1.01 1.01 Lin Mu WGFEM and Applications Oct 26-29, 2015 14 / 34

Brinkman WG Methods for Brinkman Problems Example 1. Let Ω = (0, 1) 2, u = curl(sin 2 (πx) sin 2 (πy)), and p = sin(πx). Table: Numerical Convergence Test for Crouzeix Raviart element, Raviart Thomas element, and WG element. ɛ 1 2 2 2 4 2 8 0 Crouzeix-Raviart rate H 1,velocity 0.98 0.97 0.74 0.03-0.03 rate L 2,pressure 1.00 0.93 0.98 0.12-0.03 Raviart-Thomas rate H 1,velocity -0.07-0.07 0.28 0.97 0.97 rate L 2,pressure -0.04 0.08 0.86 1.01 1.01 WG-FEM rate H 1,velocity 1.00 0.99 0.98 0.97 0.97 rate L 2, velocity 2.00 2.00 1.96 1.91 1.91 rate L 2, pressure 1.00 1.00 0.99 0.98 0.98 Lin Mu WGFEM and Applications Oct 26-29, 2015 14 / 34

Brinkman WG Methods for Brinkman Problems Next, we use the Brinkman equation in the following equivalent form for the following more practical examples: µ u + p + µκ 1 u = f, in Ω (1) Here κ denotes the permeability. Below, we shall: (1) Taking both µ and κ as variants. u = 0, in Ω (2) u = g. (3) (2) Investigate the performance of weak Galerkin algorithm to more practical problems. For all the following test problems, we have the same setting: Ω = (0, 1) (0, 1), µ = 0.01, f = 0, g = [1, 0] T. Lin Mu WGFEM and Applications Oct 26-29, 2015 15 / 34

Brinkman WG Methods for Brinkman Problems Example 2. Fluid flow in Vuggy reservoirs. Figure: (a) Profile of κ 1 for vuggy medium; (b) Pressure profile; (c) Velocity of x component; (d) Velocity of y component. Lin Mu WGFEM and Applications Oct 26-29, 2015 16 / 34

Brinkman WG Methods for Brinkman Problems Example 3. Flow through fibrous materials. Figure: (a) Profile of κ 1 for fibrous materials; (b) Pressure profile; (c) Velocity of x component; (d) Velocity of y component. Lin Mu WGFEM and Applications Oct 26-29, 2015 17 / 34

Brinkman WG Methods for Brinkman Problems Example 4. Flow through open foam. Figure: (a) Profile of κ 1 for open foam; (b) Pressure profile; (c) Velocity of x component; (d) Velocity of y component. Lin Mu WGFEM and Applications Oct 26-29, 2015 18 / 34

MsWG Application 2: Multiscale Weak Galerkin Finite Element Methods Many scientific and engineering problems involve multiple scales Difficulty of direct numerical solution: size of the computation ɛ = 0.4, P = 1.8 ɛ = 0.2, P = 1.8 ɛ = 0.1, P = 1.8 Figure: Plot of a(x/ɛ) = 1 4+P (sin(2πx/ɛ)+sin(2πy/ɛ)) Lin Mu WGFEM and Applications Oct 26-29, 2015 19 / 34

MsWG Multiscale Weak Galerkin Finite Element Methods Capture the multiscale structure of the solution via localized basis functions Basis functions contain information about the scales that are smaller than the local numerical scale (muliscale information) Let T H be a partition of Ω into coarse elements K. On each element K T H, Ψ j is the solution of the local problem: Ψ j = Φ e,j on K and Define WG multiscale vector space: a K (Ψ j, v) = 0, v V 0 h (K) V S H = span{ψ 1,, Ψ n }. Weak Galerkin multiscale method: Find Φ H V S H satisfying a(φ H, v) = (f, v), v V S H (4) Lin Mu WGFEM and Applications Oct 26-29, 2015 20 / 34

MsWG Part I: Multiscale Weak Galerkin FEM (MsWG) K K Lin Mu WGFEM and Applications Oct 26-29, 2015 21 / 34

MsWG Part I: Multiscale Weak Galerkin FEM (MsWG) K 16 15 14 K 1 2 3 13 4 5 12 6 7 8 11 10 9 Figure: Coarse Mesh VS Fine Mesh Degree of Freedom Lin Mu WGFEM and Applications Oct 26-29, 2015 21 / 34

MsWG Part I: Multiscale Weak Galerkin FEM (MsWG) K 16 15 14 K 1 2 3 13 4 5 12 6 7 8 11 10 9 Figure: Coarse Mesh VS Fine Mesh Degree of Freedom Theorem a Let Φ H and u h be the solutions of the WG multiscale method in snapshot space and WG method on the find grid respectively. Then we have u h Φ H CH f. (5) a Efendiev, Y., Mu, L., Wang, J., Ye, X. A weak Galerkin general mutiscale finite element method, in processing. Lin Mu WGFEM and Applications Oct 26-29, 2015 21 / 34

MsWG Numerical Results for MsWG Example 5. Problem is set as Ω = (0, 1) 2, a = I and u = cos(πx) cos(πy). Table: Example 5. Convergence rate for triangular mesh: Linear and quadratic weak Galerkin elements. Mesh u h Φ H order u h Φ H order Linear weak Galerkin element Level 1 7.2215e-2 2.3575e-2 Level 2 2.0804e-2 1.7954 5.9408e-3 1.9885 Level 3 5.9682e-3 1.8015 1.4881e-3 1.9972 Level 4 1.6698e-3 1.8376 3.7220e-4 1.9993 Level 5 4.5799e-4 1.8663 9.3060e-5 1.9998 Level 6 1.2382e-4 1.8871 2.3266e-5 1.9999 Quadratic weak Galerkin element Level 1 1.2218e-2 1.5319e-3 Level 2 2.1856e-3 2.4829 1.9107e-4 3.0031 Level 3 3.8978e-4 2.4873 2.3849e-5 3.0021 Level 4 6.9196e-5 2.4939 2.9786e-6 3.0012 Level 5 1.2256e-5 2.4972 3.7217e-7 3.0006 Level 6 2.1667e-6 2.4999 4.6511e-8 3.0003 Lin Mu WGFEM and Applications Oct 26-29, 2015 22 / 34

MsWG Example 6: a(x/ɛ) = 1 4+P (sin(2πx/ɛ)+sin(2πy/ɛ)) and u = 4 P 2 2 (x 2 + y 2 ). Table: Example 6. Convergence rate: Linear weak Galerkin element on triangular mesh with fixed H = 1/8. ɛ = 0.4 ɛ = 0.2 ɛ = 0.1 H/h u h Φ H order u h Φ H order u h Φ H order 1 7.960e-2 2.998e-1 2.134e-1 2 1.196e-2 2.7 8.689e-2 1.8 1.959e-1 0.12 4 1.012e-3 3.6 4.289e-3 4.3 5.518e-2 1.8 8 2.596e-4 2.0 5.943e-4 2.9 2.839e-3 4.3 16 6.494e-5 2.0 1.611e-4 1.9 3.389e-4 3.1 32 1.624e-5 2.0 4.028e-5 2.0 9.330e-5 1.9 64 4.060e-6 2.0 1.007e-5 2.0 2.333e-5 2.0 Lin Mu WGFEM and Applications Oct 26-29, 2015 23 / 34

MsWG Part II: Basis Reduction Associated each E i, an edge in coarse mesh T H, We will solve an eigenvalue problem: find λ and Z i V H (E i ) in ω(e i ) such that where a i (Z i, w) = λ i S i (Z i, w), w Θ i (6) a i (v, w) = K ω i (a w v, w w) K + h 1 v 0 v b, w 0 w b K, S i (v, w) = K ω i h 1 v 0 v b, w 0 w b K. Eigenvalues and eigenvectors: λ i 1 λ i 2 λ i J i, {Z i 1, Z i 2,, Z i J i } Lin Mu WGFEM and Applications Oct 26-29, 2015 24 / 34

MsWG Part II: Basis Reduction Define ξj i = J i m=1 zi j,m Ψ i,m, m = 1,, J i. Define VH r (E i) V H (E i ) for M i J i, VH r (E i) = span{ξ1 i, ξi 2,, ξi M i }. Let VH r be s subspace of V H, VH r = M i=1 V H r. Find u H VH r such that a(u H, v) = (f, v), v V r H. (7) Theorem Let u h and u H be the solutions of the WG method on the fine grid and the WG-GMS method respectively. The we have for k = 1 u h u H C(H f + Λ 1/2 Ψ H ), (8) where Λ = min M i=1 λi M i with M the number of E EH 0 and Ψ H is the solution of weak Galerkin multiscale methods. Lin Mu WGFEM and Applications Oct 26-29, 2015 25 / 34

MsWG Numerical Experiment of GMsWG 10 3 10 2 10 1 10 0 10 1 10 2 0 5 10 15 20 Figure: Example 7. (a). Plot of high contrast coefficient; (b). Reduction of eigenvalues; (c). Reference solution; (d). GMsWG solution. Table: Example 7. Error profiles. n = 100, N = 5 n = 100, N = 10 n = 100, N = 20 dof per E u H u h / u h u H u h / u h u H u h / u h 1 8.0463e-01 4.4399e-01 4.7785e-02 3 5.2420e-01 1.6139e-01 3.3018e-02 5 2.9955e-01 7.6311e-02 2.9196e-02 7 2.3309e-01 7.3175e-02 2.8767e-02 9 2.3153e-01 7.1989e-02 2.8051e-02 10 2.3114e-01 7.1358e-02 2.8051e-02 20 2.3011e-01 7.1358e-02-40 2.3000e-01 - - Lin Mu WGFEM and Applications Oct 26-29, 2015 26 / 34

Solution Boundedness Application 3: Solution Boundedness Example 8: (Numerical Locking) In this test, the permeability tensor is defined by 5 D = ( ) 1 0, with δ = 10 5 or 10 6. 0 δ The exact solution is taken to be u = sin(2πx)e 2π 1/δy. We consider elliptic problem with non-homogeneous Neumann boundary condition and right hand side is f = (D u). The uniqueness of solution is enforced by Ω udx = 0. Some numerical locking for finite element scheme. 5 Raphaele Herbin and Florence Hubert, Benchmark on Discretization Schemes for Anisotropic Diffusion Problems on General Grids, Finite volumes for complex applications V, Wiley, 2008. Lin Mu WGFEM and Applications Oct 26-29, 2015 27 / 34

Solution Boundedness Numerical Results for Numerical Locking Problem 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 Figure: Example 8. (a). Mesh 1; (b). Exact solution. 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 Table: Example 8. Maxmimum and Minimum values of CG and WG element on Mesh 1. Continuous element WG element δ = 1e5 Min Max Min Max -1.293e-2 1.320e-2-6.227e-1 7.035e-1 δ = 1e6 Min Max Min Max -1.548e-3 1.283e-3-5.822e-1 7.6469e-1 Lin Mu WGFEM and Applications Oct 26-29, 2015 28 / 34

Solution Boundedness Numerical Results for Numerical Locking Problem Table: Example 8. Maxmimum and Minimum values of CG and WG element on Mesh 2 and Mesh 3. Continuous element WG element δ = 1e5 Min Max Min Max Mesh 2-3.768e-2 3.816e-2-9.018e-1 9.709e-1 Mesh 3-1.137e-1 1.134e-1-9.964e-1 9.952e-1 δ = 1e6 Min Max Min Max Mesh 2-3.807e-3 3.807e-3-9.432e-1 9.474e-1 Mesh 3-1.202e-2 1.198e-2-9.950e-1 9.875e-1 Lin Mu WGFEM and Applications Oct 26-29, 2015 29 / 34

Applications Solution Boundedness Plots for Numerical Locking Problem 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0 0 0.2 0.4 0.6 0.8 1 1 0.8 0.6 0.4 0.2 0 0 Figure: Example 8. CG solution (top) and WG solution (bottom) for δ = 106 on Mesh 1, Mesh 3, and Mesh 5. Lin Mu WGFEM and Applications Oct 26-29, 2015 30 / 34

Summary Table of Contents 1 Weak Galerkin Finite Element Methods Motivation Implementation 2 Applications Brinkman Problems Multiscale Weak Galerkin Finite Element Methods Solution Boundedness 3 Summary and Future Work Summary Future Work Lin Mu WGFEM and Applications Oct 26-29, 2015 31 / 34

Summary Summary Summary The Weak Galerkin finite element methods provide a general framework for numerical simulation of partial differential equations. The Weak Galerkin Method employs discontinuous functions and can be performed on meshes with almost arbitrary shape. The degree of freedom of WG-FEM can be reduced efficiently by Schur complement. Simple formulation and easy implementation. Lin Mu WGFEM and Applications Oct 26-29, 2015 32 / 34

Summary Future Work Future Work Weak Galerkin simulation of electrostatics problems Domain decomposition preconditioner and parallel applications Hp adaptive of weak Galerkin finite element methods Lin Mu WGFEM and Applications Oct 26-29, 2015 33 / 34

Summary Future Work Thank you! Contact me at: mul1@ornl.gov Lin Mu WGFEM and Applications Oct 26-29, 2015 34 / 34