weak galerkin finite element methods and applications · weak galerkin finite element methods and...

Weak Galerkin Finite Element Methods andApplications

Lin [email protected]

Computational and Applied MathematicsComputationa Science and Mathematics Division

Oak Ridge National Laboratory

Georgia Institute of TechnologyOct 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


WGFEM

Table of Contents

1 Weak Galerkin Finite Element MethodsMotivationImplementation

2 ApplicationsBrinkman ProblemsMultiscale Weak Galerkin Finite Element MethodsSolution Boundedness

3 Summary and Future WorkSummaryFuture Work


WGFEM Motivation

Limitations of Continuous Finite Element Methods

• Lowest order of element does not work• P1 − P0 Continuous element not

stable for Stokes problem

• Difficult to construct high ordercontinuous elements.• C1 element: Argyris element,

polynomial with degree 5

• Not compatible to modern techniquesin scientific computing.• hp adaptive technique• Hybrid mesh

Figure: Argyris Finite Element

Figure: Hp adaptive FiniteElement

Figure: Hybrid mesh


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 .

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Honeycomb mesh Mesh with hanging nodes



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.



Example of WG Element: P1(K0), P0(e)

Calculating: ∇wv ∈ [P0(K)]2, withv ∈ Vh = (v0, vb) : v0|K ∈ P1(K0), vb ∈ P0(e), vb = 0 on ∂Ω

Vh = spanΦ0,i,Φb,j, i = 1, . . . , np, j = 1, . . . , ne

Φ0,i =

P1(K) on K0

0, otherwise

Φb,j =

1, on ej

0, otherwise.

V0

Vbe1

e2

e3e3

e5

e6· · ·eℓ

Figure: (a). Φ0,i is defined on K0; (b).Φb,j is defined on ∂K.

(∇wv, q)|K = −(v0,∇ · q)|K + 〈vb, q · n〉|∂K⇒ ∇wΦ0,i|K = 0, ∇wΦb,j |K =

|ej ||K|nj,K .






Φ0,i =

P1(K) on K0

0, otherwise

Φb,j =

1, on ej

0, otherwise.

V0

Vbe1

e2

e3e3

e5

e6· · ·eℓ


(∇wv, q)|K = −(v0,∇ · q)|K + 〈vb, q · n〉|∂K

⇒ ∇wΦ0,i|K = 0, ∇wΦb,j |K =|ej ||K|nj,K .






Φ0,i =

P1(K) on K0

0, otherwise

Φb,j =

1, on ej

0, otherwise.

V0

Vbe1

e2

e3e3

e5

e6· · ·eℓ


(∇wv, q)|K = −(v0,∇ · q)|K + 〈vb, q · n〉|∂K⇒ ∇wΦ0,i|K = 0, ∇wΦb,j |K =

|ej ||K|nj,K .



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.


Applications

Table of Contents





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



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).



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

ε2(∇wuh,∇wv) + (u0,v0) + s(uh,v)− (∇w · v, ph) = (f ,v),

(∇w · uh, q) = (g, q),

wheres(v,w) =

∑K∈Th

h−1K 〈v0 − vb,w0 −wb〉∂K .

4Mu, L., Wang, J., Ye, X.. A stable numerical algorithm for the Brinkman equations by weak Galerkin finite elementmethods. Journal of Computational Physics. (2014)



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

|||Qhu− uh|||+ ‖Qhp− ph‖ ≤ Chk(‖u‖k+1 + ‖p‖k),‖Q0u− u0‖ ≤ Chk+1(‖u‖k+1 + ‖p‖k).

where C is a constant independent of h and ε.



WG Methods for Brinkman Problems

Example 1. Let Ω = (0, 1)2, u = curl(sin2(πx) sin2(πy)), andp = sin(πx).

Table: Numerical Convergence Test for Crouzeix Raviart element, Raviart Thomas element, andWG element.

ε 1 2−2 2−4 2−8 0

Crouzeix-Raviart rate H1,velocity 0.98 0.97 0.74 0.03 -0.03rate L2,pressure 1.00 0.93 0.98 0.12 -0.03

Raviart-Thomas rate H1,velocity -0.07 -0.07 0.28 0.97 0.97rate L2,pressure -0.04 0.08 0.86 1.01 1.01

WG-FEM rate H1,velocity 1.00 0.99 0.98 0.97 0.97rate L2, velocity 2.00 2.00 1.96 1.91 1.91rate L2, pressure 1.00 1.00 0.99 0.98 0.98




Next, we use the Brinkman equation in the following equivalent form forthe following more practical examples:

−µ∆u +∇p+ µκ−1u = f , in Ω (1)

∇ · u = 0, in Ω (2)

u = g. (3)

Here κ denotes the permeability. Below, we shall:

(1) Taking both µ and κ as variants.

(2) Investigate the performance of weak Galerkin algorithm to morepractical 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 .




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.




Example 3. Flow through fibrous materials.

Figure: (a) Profile of κ−1 for fibrous materials; (b) Pressure profile; (c) Velocity of xcomponent; (d) Velocity of y component.




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.


Applications MsWG

Application 2: Multiscale Weak Galerkin Finite ElementMethods

• 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))


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)


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.


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.


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


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


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


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.


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 - -


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.



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


-1.548e-3 1.283e-3 -5.822e-1 7.6469e-1



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


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


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



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.


Summary

Table of Contents





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.


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


Summary Future Work

Thank you!

Contact me at: [email protected]


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