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Basic Principles of Weak Galerkin Finite Element Methods for PDEs

Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 Polytopal Element Methods in Mathematics and Engineering

Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs

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

Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs

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

Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs

Related Numerical Methods

1 FEM 2 Stabilized FEMs 3 MFD 4 DG, HDG 5 VEM Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs

Second Order Elliptic Problems

Find u ∈ H1

0(Ω) such that

(a∇u, ∇v) = (f , v), ∀v ∈ H1

0(Ω).

Procedures in the standard Galerkin finite element method:

1 Partition Ω into triangles or tetrahedra. 2 Construct a subspace, denoted by Sh ⊂ H1

0(Ω), using

piecewise polynomials.

3 Seek for a finite element solution uh from Sh such that

(a∇uh, ∇v) = (f , v) ∀v ∈ Sh.

Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs

An Out-of-Box Thinking

Replace uh and v by any distribution, and ∇uh and ∇v by another distribution, say ∇wv as the generalized derivative, and seek for a distribution uh such that (a∇wuh, ∇wv) = (f , v), ∀v ∈ Sh. Main Issues:

1 Functions in Sh 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.

Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs

Motivation for WG

The classical gradient ∇u for u ∈ C 1(K) can be computed as

• K

∇u · φ = −

• K

u ∇ · φ +

• ∂K

u(φ · n) for all φ ∈ [C 1(K)]2. The integrals on the right-hand side requires only u0 = u in the interior of K, plus ub = u (trace) on the boundary ∂K. We symbolically have

• K

∇wu · φ = −

• K

u0 ∇ · φ +

• ∂K

ubφ · n Thus, u can be extended to {u0, ub} with ∇u being extended to ∇wu.

Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs

Generalized Weak Derivatives: Foundation of WG

Weak Derivative For any u = {u0; ub} with u0 ∈ L2(K) and ub ∈ L2(∂K), the generalized weak derivative of u in the direction ν is the following linear functional on H1(K): ∂νu, φ = −

• K

u0∂νφ +

• ∂K

(n · ν)ubφ. for all φ ∈ H1(K). The generalized weak derivative shall be called weak derivative.

Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs

Weak Functions

Weak Functions A weak function on the region K refers to a generalized function v = {v0, vb} such that v0 ∈ L2(K) and vb ∈ L2(∂K). The first component v0 represents the value of v in the interior of K, and the second component vb represents v on the boundary of K. vb may or may not be related to the trace of v0 on ∂K. The space of weak functions: W (K) = {v = {v0, vb} : v0 ∈ L2(K), vb ∈ L2(∂K)}.

Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs

Weak Gradient

For any v ∈ W (K), the weak gradient of v is defined as a bounded linear functional ∇wv in H1(K) whose action on each q ∈ H1(K) is given by ∇wv, qK := −

• K

v0∇ · qdK +

• ∂K

vbq · nds, 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).

Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs

Discrete Weak Gradients

For computational purpose, the weak gradient needs to be approximated, which leads to discrete weak gradients, ∇w,r, given by

• K

∇w,rv · qdK = −

• K

v0∇ · qdK +

• ∂K

vbq · nds, for all q ∈ V (K, r). Here V (K, r) [Pr(K)]2 is a subspace. Pr(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.

Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs

Weak Finite Element Spaces

Th: polygonal/polytopal partition of the domain Ω, shape regular construct local discrete elements Wk(T) := {v = {v0, vb} : v0 ∈ Pk(T), vb ∈ Pk−1(∂T)} . patch local elements together to get a global space Sh := {v = {v0, vb} : {v0, vb}|T ∈ Wk(T), ∀T ∈ Th} . Weak FE Space Weak finite element space with homogeneous boundary value: S0

h := {v = {v0, vb} ∈ Sh, vb|∂T∩∂Ω = 0, ∀T ∈ Th} .

Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs

Weak Finite Element Functions

Element Shape Functions for P1(K)/P0(∂K): φi = {λi, 0}, i = 1, 2, 3, φ3+j = {0, τj}, j = 1, 2, · · · , N, where N is the number of sides.

Figure: WG element

Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs

Shape Regularity for Polytopal Elements

xe Ae A B C D E F ne Why Shape Regularity? The shape regularity is needed for (1) trace inequality, (2) inverse inequality, and (3) domain inverse inequality.

Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs

Weak Galerkin Finite Element Formulation

WG-FEM Find uh = {u0; ub} ∈ S0

h such that

(a∇wuh, ∇wv) + s(uh, v) = (f , v0), ∀v = {v0; vb} ∈ S0

h,

where

1 ∇wv ∈ Pk−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.

Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs

Depiction of the General WG Element

Pj−1(e) Pj−1(e) Pj−1(e) Pj(T) The polynomial spaces Pj−1(T) or Pj(T) can be used for the computation of the weak gradient ∇w.

Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs

Stabilizer

Commonly used stabilizer: s(w, v) = ρ

• T∈Th

h−1

T Qbw0 − wb, Qbv0 − vb∂T,

where Qb is the L2 projection onto Pk−1(e), e ⊂ ∂T. Discrete and computation-friendly stabilizer: s(w, v) = ρ

• T∈Th
• xj

(Qbw0 − wb)(xj) (Qbv0 − vb)(xj), where {xj} is a set of (nodal) points on ∂T.

Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs

WG from the Minimization Perspective

The original problem can be characterized as u = arg min

v∈H1

0(Ω)

1 2(a∇v, ∇v) − (f , v)

• Weak Galerkin finite element scheme

uh = arg min

v∈S0

h

1 2(a∇wv, ∇wv) − (f , v0) + 1 2s(v, v)

• .

Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs

L2 Projections and Weak Gradients

The following is a commutative diagram: H1(T) [L2(T)]d Sh(T) V (r, T) ∇ Qh Qh ∇w

• r equivalently

∇w(Qhu) = Qh(∇u), ∀u ∈ H1(T). Implication: The discrete weak gradient is a good approximation of the true gradient operator.

Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs

Error Estimate

With the correct regularity assumptions, one has the following

• ptimal order error estimate

Qhu − u00 + hQhu − uh1,h ≤ Chk+1uk+1. Error estimates in negative norms hold true as well. Thus, the WG-FEM solutions are of superconvergent at certain places.

Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs

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 Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs

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 H1, H(div), H(curl), H2, or weighted-version of them. Assume Vh: finite dimensional spaces that approximate V ah(·, ·): bilinear forms on Vh × Vh that approximate a(·, ·) fh: linear functionals on Vh that approximate f . sh(·, ·): stabilizers that provide necessary “smoothness” In WG for Poisson equation, the first bilinear form refers to ah(u, v) = (a∇wu, ∇wv), and the second one sh(·, ·) is the stabilizer.

Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs

Abstract WG

Abstract WG Find uh ∈ Vh such that ah(uh, v) + sh(uh, v) = fh(v), ∀ v ∈ Vh. Some Assumptions: Regularity: The solution of the abstract problem lies in a subspace H ⊂ V The (discrete) norm · Vh can be extended to H + Vh so that the topology of H is given by the family of semi-norms · Vh, h ∈ (0, h0) Boundedness and Coercivity: The bilinear form awh(·, ·) := ah(·, ·) + sh(·, ·) is bounded and coercive in Vh.

Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs

Almost Consistency

The Abstract WG algorithm is said to be almost consistent if there exists a linear projection/interpolation operator Qh : V − → Vh such that for each uf ∈ H of the abstract problem, one has Interpolation approximation: lim

h→0 uf − Qhuf Vh = 0

Residual consistency: lim

h→0

sup

v=0,v∈Vh

|ah(Qhuf , v) − fh(v)| vVh = 0 Almost smoothness: lim

h→0

sup

v=0,v∈Vh

|sh(Qhuf , v)| vVh = 0.

Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs

Convergence

Convergence Assume that the Abstract WG algorithm is almost consistent. Furthermore, assume that awh(·, ·) is bounded and coercive in Vh, and the solution to the abstract problem lies in the subspace H. Then, we have lim

h→0 uf − uhVh = 0.

For the second order elliptic problem, the convergence can be interpreted as lim

h→0

• Qh(∇uf ) − ∇wuh0 + s(uh, uh)

1 2

• = 0.

Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs

WG: a New Paradigm of Discretization

Primal Formulation Find uh such that (a∇wuh, ∇wv) + s(uh, v) = (f , v), ∀ v. Key in WG: discrete weak gradient + stabilization to ensure a weak continuity of uh in H1. Primal-Mixed Formulation Find uh and qh such that (a−1qh, v) + (∇wuh, v) = 0, ∀ v s(uh, w) − (qh, ∇ww) = (f , w), ∀ w. Key in WG: discrete weak gradient + stabilization to provide a weak continuity for uh in H1.

Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs

WG: a New Paradigm of Discretization

Dual-Mixed Formulation Find qh and uh such that s(qh, vh) + (a−1qh, v) − (u, ∇w · v) = 0, ∀ v (∇w · qh, 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 Pj+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.

Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs

WG Can Be Hybridized

Hybridized Dual-Mixed Formulation Find qh, uh, and λh such that s(qh, vh) + (a−1qh, v) − (u, ∇w · v) +

• T

λh, vb · n∂T = 0, (∇w · qh, w) +

• T

σ, qb · n∂T = (f , w). Key in HWG: variable reduction in the sense that qh and uh can be eliminated locally on each element. Pjn Pjn Pjn [Pj]d × Pj+1

Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs

Linear Elasticity Problems

Model Problem: Find a displacement vector field u satisfying − ∇ · σ(u) = f, in Ω, u =

• u,
• n Γ.

Stress-strain relation for linear, homogeneous, and isotropic materials: σ(u) = 2µε(u) + λ(∇ · u)I, (Primal Form) Find u ∈ [H1(Ω)]d satisfying u = u on Γ and 2(µε(u), ε(v)) + (λ∇ · u, ∇ · v) = (f, v), ∀v ∈ [H1

0(Ω)]d.

Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs

Linear Elasticity Problems-Mixed Formulation

Introducing a pressure variable p = λ∇ · u, the elasticity problem can be reformulated as follows: (Mixed Formulation) Find u ∈ [H1(Ω)]d and p ∈ L2(Ω) satisfying u = u on Γ, the compatibility condition

• Ω λ−1pdx =
• Γ

u · nds, 2(µε(u), ε(v)) + (∇ · v, p) = (f, v), ∀v ∈ [H1

0(Ω)]d,

(∇ · u, q) − (λ−1p, q) = 0, ∀q ∈ L2

0(Ω).

Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs

Weak Divergence Operator

The space of weak vector-valued functions in K V (K) = {v = {v0, vb} : v0 ∈ [L2(K)]d, vb ∈ [L2(∂K)]d}. Weak Divergence The weak divergence of v ∈ V (K), ∇w · v, is a bounded linear functional on H1(K), so that its action on any φ ∈ H1(K) is given by ∇w · v, φK := −(v0, ∇φ)K + vb · 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 = −(v0, ∇φ)K + vb · n, φ∂K, ∀φ ∈ Pr(K).

Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs

Weak Finite Element Spaces

[Pk−1]2 + PRM [Pk−1]2 + PRM [Pk−1]2 + PRM [Pk]2 ∇wv ∈ [Pk−1(T)]2×2 ∇w · v ∈ Pk−1(T).

Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs

Weak Galerkin Algorithms for Primal Formulation

WG-FEM Primal Find uh = {u0, ub} ∈ Vh with ub = Qbˆ u on Γ such that for all v = {v0, vb} ∈ V 0

h ,

• T∈Th

2(µεw(uh), εw(v))T + (λ∇w · uh, ∇w · v)T + s(uh, v) = (f, v0). Qb: L2 projection onto [Pk−1(e)]d + PRM(e) εw(u) = 1

2(∇wu + ∇wuT)

Stablizer: s(w, v) =

T∈Th h−1 T Qbw0 − wb, Qbv0 − vb∂T

Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs

Weak Galerkin Algorithms for Mixed Formulation

WG-FEM in Mixed Form Find uh = {u0, ub} ∈ Vh and ph ∈ Wh satisfying ub = Qbˆ u on Γ, the compatibility condition (λ−1ph, 1) =

• Γ

u · nds, and 2(µεw(uh), εw(v))h + s(uh, v) + (∇w · v, ph)h = (f, v0), ∀v ∈ V 0

h ,

(∇w · uh, q)h − λ−1(ph, q) = 0, ∀q ∈ W 0

h .

Wh = {q : q|T ∈ Pk−1(T), T ∈ Th} W 0

h = Wh ∩ L2 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.

Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs

Error Estimate in a Discrete H1-Norm

Error Estimates and Convergence in H1 Let the exact solution be sufficiently smooth such that (u; p) ∈ [Hk+1(Ω)]d × Hk(Ω). Let (uh; ph) ∈ Vh × Wh be the weak Galerkin finite element solution. |||Qhu−uh|||+λ− 1

2 Qhp−ph+|||Qhp−ph|||0 ≤ Chk(uk+1+pk),

where C is a generic constant independent of (u; p). Consequently, |||u − uh||| + λ− 1

2 p − ph + |||p − ph|||0 ≤ Chk(uk+1 + pk). Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs

Error Estimate in L2-Norm

Error Estimates and Convergence in L2 Assume that the exact solution is sufficiently smooth such that (u; p) ∈ [Hk+1(Ω)]d × Hk(Ω). Let (uh; ph) ∈ Vh × Wh be the weak Galerkin finite element solution. Then, under the regularity assumption, there exists a constant C, such that Q0u − u0 ≤ Chk+s uk+1 + pk

• .

Moreover, u − u0 ≤ Chk+s uk+1 + pk

• .

Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs

Numerical Results

Ω = (0, 1)2 the exact solution u = sin(x) sin(y) 1

• Table: WG based on {P1(T)/PRM(e)}, λ = 1, µ = 0.5.

1/h u0 − Q0u

• rder

ub − Qbu

• rder

|||uh − Qhu|||

• rder

2 0.0750 – 0.0424 – 0.3103 – 4 0.0192 1.97 0.0115 1.88 0.1566 0.99 8 0.0049 1.98 0.0031 1.87 0.0787 0.99 16 0.0012 1.99 0.0008 1.93 0.0394 1.00 32 0.0003 2.00 0.0002 1.97 0.0197 1.00

Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs

Numerical Results

Table: WG based on {P1(T)/P1(e)}, λ = 1, µ = 0.5.

1 h

uh − Q0u

• rder

ub − Qbu

• rder

|||uh − Qhu|||

• rder

2 0.0743 – 0.0424 – 0.3082 – 4 0.0190 1.96 0.0113 1.90 0.1555 0.99 8 0.0048 1.98 0.0031 1.88 0.0782 0.99 16 0.0012 1.99 0.0008 1.93 0.0392 1.00 32 0.0003 2.00 0.0002 1.97 0.0196 1.00

Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs

Numerical Results: Locking-Free Experiments

Ω = (0, 1)2 the exact solution u = sin(x) sin(y) cos(x) cos(y)

• + λ−1

x y

• Table: WG based on {P1(T)/PRM(e)}, µ = 0.5, and λ = 1.

1 h

uh − Q0u

• rder

ub − Qbu

• rder

|||uh − Qhu|||

• rder

2 0.0352 – 0.0331 – 0.1544 – 4 0.0097 1.86 0.0120 1.46 0.0834 0.89 8 0.0026 1.91 0.0037 1.68 0.0433 0.94 16 0.0007 1.96 0.0010 1.87 0.0220 0.98 32 0.0002 1.98 0.0003 1.96 0.0110 0.99

Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs

Numerical Results: Locking-Free Experiments

Table: WG based on {P1(T)/PRM(e)}, µ = 0.5, and λ = 1, 000, 000.

1 h

uh − Q0u

• rder

ub − Qbu

• rder

|||uh − Qhu|||

• rder

2 0.0344 – 0.0290 – 0.1447 – 4 0.0100 1.79 0.0113 1.36 0.0773 0.90 8 0.0028 1.82 0.0038 1.59 0.0403 0.94 16 0.0008 1.90 0.0011 1.81 0.0205 0.97 32 0.0002 1.96 0.0003 1.93 0.0103 0.99

Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs

Numerical Results: Locking-Free Experiments

Table: WG based on {P1(T)/P1(e)}, µ = 0.5, and λ = 1.

1 h

uh − Q0u

• rder

ub − Qbu

• rder

|||uh − Qhu|||

• rder

2 0.0341 – 0.0313 – 0.1518 – 4 0.0093 1.87 0.0115 1.45 0.0816 0.90 8 0.0025 1.91 0.0036 1.67 0.0424 0.95 16 0.0006 1.96 0.0010 1.86 0.0215 0.98 32 0.0002 1.98 0.0003 1.95 0.0108 0.99

Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs

Numerical Results: Locking-Free Experiments

Table: WG based on {P1(T)/P1(e)}, µ = 0.5, and λ = 1, 000, 000.

1 h

uh − Q0u

• rder

ub − Qbu

• rder

|||uh − Qhu|||

• rder

2 0.0340 – 0.0280 – 0.1439 – 4 0.0098 1.79 0.0111 1.34 0.0768 0.91 8 0.0028 1.82 0.0037 1.58 0.0400 0.94 16 0.0007 1.90 0.0011 1.81 0.0203 0.97 32 0.0002 1.96 0.0003 1.93 0.0102 0.99

Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs

The Fokker-Planck Equation

describe the time evolution of the probability density function

• f the velocity of a particle under the influence of drag forces

and random forces, as in Brownian motion. assume the stochastic differential equation: dXt = µ(Xt, t)dt + σ(Xt, t)dWt the probability density f (x, t) for the random vector Xt satisfies the Fokker-Planck equation ∂f ∂t + ∇ · (µf ) − 1 2

N

• i,j=1

∂2

ij[Dijf ] = 0,

where µ = (µ1, · · · , µN) is the drift vector and Dij(x, t) =

M

• k=1

σik(x, t)σjk(x, t) is the diffusion tensor.

Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs

Model Problem

Find u = u(x) satisfying

d

• i,j=1

∂2

ij(aiju) =g,

in Ω, u =0,

• n ∂Ω.

assume that a(x) is non-smooth, weak formulation is given by seeking u such that

d

• i,j=1

(u, aij∂2

ijw) = (g, w),

∀v ∈ H2(Ω) ∩ H1

0(Ω).

Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs

A Cauchy Problem for Elliptic Equations

The model problem seeks u such that −∆u = f , in Ω, u = 0,

• n Γ ⊂ ∂Ω,

∂u ∂n = ψ,

• n Γ ⊂ ∂Ω.

This is usually an ill-posed problem which does not have a solution

• r has many solutions. Let Γc = ∂Ω/Γ. A variational form for this

problem seeks u ∈ H1

0,Γ(Ω) such that

(∇u, ∇w) = (f , w) + ψ, wΓ, for all w ∈ H1

0,Γc(Ω).

Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs

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 L2, H1, H(div), H(curl), H2, 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.

Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs

Specific Examples (Revisited)

For the Fokker-Planck, we have V = L2 and W = H2 ∩ H1 and b(v, w) :=

d

• i,j=1

(v, aij∂2

ijw).

For the Cauchy problem for Poisson equation, V × W = H1

0,Γ(Ω) × H1 0,Γc(Ω) and

b(v, w) := (∇v, ∇w). Note that the inf-sup condition may not be satisfied.

Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs

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 uh ∈ Vh and λh ∈ Wh such that s1(uh, v) − bh(v, λh) = 0, ∀v ∈ Vh s2(λh, w) + bh(uh, w) = fh(w), ∀ w ∈ Wh. s1(·, ·): stabilizer/smoother in Vh s2(·, ·): stabilizer/smoother in Wh

Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs

The Primal-Dual WG for Elliptic Cauchy Problem

the bh(·, ·)-form is given by bh(v, w) := (∇wv, ∇ww), Both stabilizers are given by: s(u, v) =

• T∈Th

h−1

T u0−ub, v0−vb∂T+hT∂nu0−ugn, ∂nv0−vgn∂T.

Error estimates and numerical experiments are on the way.

Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs

Numerical Tests

the right-hand side f is given by 2a11+2a22−10a12−10a21−50 sin(30(x −0.5)2+30(y −0.5)2) the boundary condition u = x2 + 2y2 − 5xy on ∂Ω Ω = (0, 1)2

Figure: WG finite element solution with coefficients a11 = 3, a12 = a21 = 1 and a22 = 2 in mesh size 1.250e-01 (ρh is piecewise linear function).

Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs

Numerical Tests

Figure: WG finite element solution with coefficients a11 = 10, a12 = a21 = (0.25x)2(0.25y)2, a22 = 10 in mesh size 1.250e-01 (ρh is piecewise linear function).

Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs

Numerical Tests

Figure: WG finite element solution with coefficients a11 = 3, a12 = a21 = 1 and a22 = 2 in mesh size 1.250e-01 (ρh is a piecewise constant).

Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs

Numerical Tests

Figure: WG finite element solution with coefficients a11 = 10, a12 = a21 = (0.25x)2(0.25y)2, a22 = 10 in mesh size 1.250e-01 (ρh is a piecewise constant).

Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs

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. Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs

Thanks for your attention!

Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt] Basic Principles of Weak Galerkin Finite Element Methods for PDEs