Weak Galerkin Finite Element Methods and Applications Lin Mu [email protected]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
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Weak Galerkin Finite Element Methods andApplications
Lin Mu WGFEM and Applications Oct 26-29, 2015 5 / 34
WGFEM Implementation
Weak Galerkin Formulation
The weak form of the PDE: seeking u ∈ H10 (Ω) satisfying:
(a∇u,∇v) = (f, v), ∀v ∈ H10 (Ω).
Weak Galerkin finite element method: seeking uh ∈ Vh satisfying:
(a∇wuh,∇wv) + s(uh, v) = (f, v), ∀v ∈ Vh,
where s(uh, v) =∑
K h−1K 〈u0 − ub, v0 − vb〉∂K .
Like traditional FEM (CG) to compute (∇uh,∇v), for WG we need tocompute:1
(a∇wuh,∇wv) =∑K
(a∇wuh,∇wv)K .
1L. Mu, J. Wang, and X. Ye. Weak Galerkin Finite Element Methods on Polytopal Meshes. International Journal ofNumerical Analysis and Modeling. 12 (2015), 31-53.
Lin Mu WGFEM and Applications Oct 26-29, 2015 6 / 34
Lin Mu WGFEM and Applications Oct 26-29, 2015 7 / 34
WGFEM Implementation
Weak Galerkin Numerical Schemes
Weak Galerkin finite element method for Stokes equations:2 seeking(uh, ph) ∈ Vh ×Wh such that for all (v, q) ∈ Vh ×Wh
(∇wuh,∇wv) + s(uh,v)− (∇w · v, ph) = (f ,v)
(∇w · uh, q) = 0.
Weak Galerkin finite element method for Biharmonic equations:3
seeking uh ∈ Vh satisfying
(∆wuh,∆wv) + s(uh, v) = (f, v), ∀v ∈ Vh.
2J. Wang, and X. Ye. A Weak Galerkin Finite Element Method for the Stokes Equations, arXiv: 1302.2707.3L. 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
Applications
Table of Contents
1 Weak Galerkin Finite Element MethodsMotivationImplementation
2 ApplicationsBrinkman ProblemsMultiscale Weak Galerkin Finite Element MethodsSolution Boundedness
3 Summary and Future WorkSummaryFuture Work
Lin Mu WGFEM and Applications Oct 26-29, 2015 9 / 34
Applications Brinkman
Application 1: Brinkman Problems
Brinkman equations:
−ε2∆u + u +∇p = f , in Ω,
∇ · u = g, in Ω
where 0 ≤ ε2 ≤ 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
Applications Brinkman
Challenge of algorithm design for the Brinkman equations
• The main challenge for solving Brinkman equations is in theconstruction of numerical schemes that are stable for both the Darcyflow and the Stokes flow.
• Stokes stable elements such as Crouzeix-Raviart element, MINIelement and Taylor-Hood element do not work well for the Darcy flow(small ε).
• Darcy stable elements such as RT elements and BDM elements donot work well for the Stokes flow (large ε).
Lin Mu WGFEM and Applications Oct 26-29, 2015 11 / 34
Applications Brinkman
WG methods for the Brinkman equations
The weak form of the Brinkman equations: find(u, p) ∈ H1
0 (Ω)d × L20(Ω) s.t. for (v, q) ∈ H1
0 (Ω)d × L20(Ω)
ε2(∇u,∇v) + (u,v)− (∇ · v, p) = (f ,v),
(∇ · u, q) = (g, q).
Weak Galerkin finite element methods4 for the Brinkman equations:find (uh, ph) ∈ Vh ×Wh s.t. for (v, q) ∈ Vh ×Wh
4Mu, L., Wang, J., Ye, X.. A stable numerical algorithm for the Brinkman equations by weak Galerkin finite elementmethods. Journal of Computational Physics. (2014)
Lin Mu WGFEM and Applications Oct 26-29, 2015 12 / 34
Applications Brinkman
Uniform convergence rate of WG method
Theorem (Mu, Wang, Ye, 2014, Uniform convergence rate)
Let (u; p) ∈ (H10 (Ω) ∩Hk+1(Ω))d × L2
0(Ω) ∩Hk(Ω) with k ≥ 1 be thesolution of the Brinkman equations and (uh; ph) ∈ Vh ×Wh be thesolutions of the weak Galerkin methods. Then
• Many scientific and engineering problems involve multiple scales
• Difficulty of direct numerical solution: size of the computation
ε = 0.4, P = 1.8 ε = 0.2, P = 1.8 ε = 0.1, P = 1.8
Figure: Plot of a(x/ε) = 14+P (sin(2πx/ε)+sin(2πy/ε))
Lin Mu WGFEM and Applications Oct 26-29, 2015 19 / 34
Applications MsWG
Multiscale Weak Galerkin Finite Element Methods
• Capture the multiscale structure of the solution via localized basisfunctions
• Basis functions contain information about the scales that are smallerthan the local numerical scale (muliscale information)
Let TH be a partition of Ω into coarse elements K. On each elementK ∈ TH , Ψj is the solution of the local problem: Ψj = Φe,j on ∂K and
aK(Ψj , v) = 0, ∀v ∈ V 0h (K)
Define WG multiscale vector space:
V SH = spanΨ1, · · · ,Ψn.
Weak Galerkin multiscale method: Find ΦH ∈ V SH satisfying
a(ΦH , v) = (f, v),∀v ∈ V SH (4)
Lin Mu WGFEM and Applications Oct 26-29, 2015 20 / 34
Applications MsWG
Part I: Multiscale Weak Galerkin FEM (MsWG)
K
K
1 2 3 4 5 6 7 8
9
10
11
12
13
14
15
16
Figure: Coarse Mesh VS Fine Mesh Degree of Freedom
Theorema Let ΦH and uh be the solutions of the WG multiscale method insnapshot space and WG method on the find grid respectively. Then wehave
|||uh − ΦH ||| ≤ CH‖f‖. (5)
aEfendiev, 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
Applications MsWG
Part I: Multiscale Weak Galerkin FEM (MsWG)
K
K 1 2 3 4 5 6 7 8
9
10
11
12
13
14
15
16
Figure: Coarse Mesh VS Fine Mesh Degree of Freedom
Theorema Let ΦH and uh be the solutions of the WG multiscale method insnapshot space and WG method on the find grid respectively. Then wehave
|||uh − ΦH ||| ≤ CH‖f‖. (5)
aEfendiev, 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
Applications MsWG
Part I: Multiscale Weak Galerkin FEM (MsWG)
K
K 1 2 3 4 5 6 7 8
9
10
11
12
13
14
15
16
Figure: Coarse Mesh VS Fine Mesh Degree of Freedom
Theorema Let ΦH and uh be the solutions of the WG multiscale method insnapshot space and WG method on the find grid respectively. Then wehave
|||uh − ΦH ||| ≤ CH‖f‖. (5)
aEfendiev, 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
Applications 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 Galerkinelements.
Mesh |||uh − ΦH ||| order ‖uh − ΦH‖ 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
Applications MsWG
Example 6: a(x/ε) = 14+P (sin(2πx/ε)+sin(2πy/ε)) and u =
√4−P 2
2 (x2 + y2).
Table: Example 6. Convergence rate: Linear weak Galerkin element on triangular mesh withfixed H = 1/8.
ε = 0.4 ε = 0.2 ε = 0.1
H/h ‖uh − ΦH‖ order ‖uh − ΦH‖ order ‖uh − ΦH‖ 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
Applications MsWG
Part II: Basis Reduction
Associated each Ei, an edge in coarse mesh TH , We will solve aneigenvalue problem: find λ and Zi ∈ VH(Ei) in ω(Ei) such that
ai(Zi, w) = λiSi(Zi, w), ∀w ∈ Θi (6)
where
ai(v, w) =∑K∈ωi
(a∇wv,∇ww)K + h−1〈v0 − vb, w0 − wb〉∂K ,
Si(v, w) =∑K∈ωi
h−1〈v0 − vb, w0 − wb〉∂K .
Eigenvalues and eigenvectors:
λi1 ≤ λi2 ≤ · · · ≤ λiJi ,Zi1, Zi2, · · · , ZiJi
Lin Mu WGFEM and Applications Oct 26-29, 2015 24 / 34
Applications MsWG
Part II: Basis Reduction
Define ξij =∑Ji
m=1 zij,mΨi,m, m = 1, · · · , Ji. Define V r
H(Ei) ⊂ VH(Ei) for
Mi ≤ Ji, V rH(Ei) = spanξi1, ξi2, · · · , ξiMi
. Let V rH be s subspace of VH ,
VrH = ∪Mi=1VrH .
Find uH ∈ V rH such that
a(uH , v) = (f, v), ∀v ∈ V rH . (7)
Theorem
Let uh and uH be the solutions of the WG method on the fine grid andthe WG-GMS method respectively. The we have for k = 1
|||uh − uH ||| ≤ C(H‖f‖+ Λ−1/2|||ΨH |||), (8)
where Λ = minMi=1 λiMi
with M the number of E ∈ E0H and ΨH is the
solution of weak Galerkin multiscale methods.
Lin Mu WGFEM and Applications Oct 26-29, 2015 25 / 34
Applications MsWG
Numerical Experiment of GMsWG
0 5 10 15 2010
−2
10−1
100
101
102
103
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 = 20dof per E ‖uH − uh‖/‖uh‖ ‖uH − uh‖/‖uh‖ ‖uH − uh‖/‖uh‖
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
Applications Solution Boundedness
Application 3: Solution Boundedness
Example 8: (Numerical Locking) In this test, the permeability tensor isdefined by5
D =
(1 00 δ
), with δ = 105 or 106.
The exact solution is taken to be u = sin(2πx)e−2π√
1/δy. We considerelliptic problem with non-homogeneous Neumann boundary condition andright hand side is f = −∇ · (D∇u). The uniqueness of solution is enforcedby∫
Ω udx = 0.Some numerical locking for finite element scheme.
5Raphaele Herbin and Florence Hubert, Benchmark on Discretization Schemes for Anisotropic Diffusion Problems on GeneralGrids, Finite volumes for complex applications V, Wiley, 2008.
Lin Mu WGFEM and Applications Oct 26-29, 2015 27 / 34
Applications Solution Boundedness
Numerical Results for Numerical Locking Problem
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
Figure: Example 8. (a). Mesh 1; (b). Exact solution.
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
Applications Solution Boundedness
Numerical Results for Numerical Locking Problem
Table: Example 8. Maxmimum and Minimum values of CG and WG element on Mesh 2 andMesh 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
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
Figure: Example 8. CG solution (top) and WG solution (bottom) for δ = 106 on Mesh 1, Mesh3, and Mesh 5.
Lin Mu WGFEM and Applications Oct 26-29, 2015 30 / 34
Summary
Table of Contents
1 Weak Galerkin Finite Element MethodsMotivationImplementation
2 ApplicationsBrinkman ProblemsMultiscale Weak Galerkin Finite Element MethodsSolution Boundedness
3 Summary and Future WorkSummaryFuture Work
Lin Mu WGFEM and Applications Oct 26-29, 2015 31 / 34
Summary Summary
Summary
• The Weak Galerkin finite element methods provide a generalframework for numerical simulation of partial differential equations.
• The Weak Galerkin Method employs discontinuous functions and canbe performed on meshes with almost arbitrary shape.
• The degree of freedom of WG-FEM can be reduced efficiently bySchur 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