basic principles of weak galerkin finite element methods for pdes
TRANSCRIPT
Basic Principles of Weak Galerkin FiniteElement Methods for PDEs
Junping WangComputational Mathematics
Division of Mathematical SciencesNational 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.org2 Partial List of Contributors:
Xiu Ye, University of ArkansasChunmei Wang, Georgia Institute of TechnologyLin Mu, Michigan State UniversityGuowei Wei, Michigan State UniversityYanqiu Wang, Oklahoma State UniversityLong Chen, University of California, IrvineShan Zhao, University of AlabamaRan Zhang, Jilin University, ChinaRuishu Wang, Jilin University, ChinaQilong 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 gradientstabilization (weak continuity)implementation and error analysis
2 An Abstract Framework3 WG-FEM for Model PDEs
mixed formulationhybridized WGlinear elasticity
4 Primal-Dual Weak Galerkin – What is it briefly?
The Fokker-Planck equationThe Cauchy problem for elliptic equationsAn 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 ∈ H10 (Ω) such that
(a∇u,∇v) = (f , v), ∀v ∈ H10 (Ω).
Procedures in the standard Galerkin finite element method:
1 Partition Ω into triangles or tetrahedra.
2 Construct a subspace, denoted by Sh ⊂ H10 (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 anotherdistribution, say ∇wv as the generalized derivative, and seek for adistribution uh such that
(a∇wuh,∇wv) = (f , v), ∀v ∈ Sh.
Main Issues:
1 Functions in Sh are to be more general (as distributions orgeneralized 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 inthe 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 extendedto ∇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 ), thegeneralized weak derivative of u in the direction ν is the followinglinear functional on H1(K ):
〈∂νu, φ〉 = −∫
Ku0∂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 functionv = v0, vb such that v0 ∈ L2(K ) and vb ∈ L2(∂K ).
The first component v0 represents the value of v in theinterior of K , and the second component vb represents v onthe 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 abounded linear functional ∇wv in H1(K ) whose action oneach q ∈ H1(K ) is given by
〈∇wv , q〉K := −∫
Kv0∇ · qdK +
∫∂K
vbq · nds,
where n is the outward normal direction on ∂K .
The weak gradient is identical with the strong gradient forsmooth weak functions (e.g., as restriction of smoothfunctions).
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 beapproximated, which leads to discrete weak gradients, ∇w ,r , givenby ∫
K∇w ,rv · qdK = −
∫K
v0∇ · qdK +
∫∂K
vbq · nds,
for all q ∈ V (K , r). Here
V (K , r) j [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 indiscretization.
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, shaperegular
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:
S0h := 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
AeA
B
CD
EF
ne
Why Shape Regularity?The shape regularity is needed for (1) trace inequality, (2) inverseinequality, 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 ∈ S0h such that
(a∇wuh, ∇wv) + s(uh, v) = (f , v0), ∀v = v0; vb ∈ S0h ,
where
1 ∇wv ∈ Pk−1(T ) is the discrete weak gradient computedlocally on each element,
2 s(·, ·) is a stabilizer enforcing a weak continuity,
3 the stabilizer s(·, ·) measures the discontinuity of the finiteelement 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 thecomputation 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−1T 〈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 minv∈H1
0 (Ω)
(1
2(a∇v ,∇v)− (f , v)
)Weak Galerkin finite element scheme
uh = arg minv∈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 ) 0
∇
Qh Qh∇w
or equivalently
∇w (Qhu) = Qh(∇u), ∀u ∈ H1(T ).
Implication:
The discrete weak gradient is a good approximation of thetrue 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 followingoptimal order error estimate
‖Qhu − u0‖0 + h‖Qhu − uh‖1,h ≤ Chk+1‖u‖k+1.
Error estimates in negative norms hold true as well. Thus, theWG-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 elementstiffness matrices
2 WG preserves physical quantities of importance: massconservation, energy conservation, etc
3 Suitable for parallel computation; element degree of freedomscan 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, orweighted-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 asubspace H ⊂ V
The (discrete) norm ‖ · ‖Vhcan 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 thereexists a linear projection/interpolation operator Qh : V −→ Vh
such that for each uf ∈ H of the abstract problem, one has
Interpolation approximation:
limh→0
‖uf − Qhuf ‖Vh= 0
Residual consistency:
limh→0
supv 6=0,v∈Vh
|ah(Qhuf , v)− fh(v)|‖v‖Vh
= 0
Almost smoothness:
limh→0
supv 6=0,v∈Vh
|sh(Qhuf , v)|‖v‖Vh
= 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
limh→0
‖uf − uh‖Vh= 0.
For the second order elliptic problem, the convergence can beinterpreted as
limh→0
(‖Qh(∇uf )−∇wuh‖0 + s(uh, uh)
12
)= 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 aweak 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 aweak 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 toensure 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 forpressure
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 canbe 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, on Γ.
Stress-strain relation for linear, homogeneous, and isotropicmaterials:
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 ∈ [H10 (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 problemcan be reformulated as follows:
(Mixed Formulation) Find u ∈ [H1(Ω)]d and p ∈ L2(Ω) satisfyingu = u on Γ, the compatibility condition
∫Ω λ
−1pdx =∫Γ u · nds,
2(µε(u), ε(v)) + (∇ · v, p) = (f, v), ∀v ∈ [H10 (Ω)]d ,
(∇ · u, q)− (λ−1p, q) = 0, ∀q ∈ L20(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 linearfunctional on H1(K ), so that its action on any φ ∈ H1(K ) is givenby
〈∇w · v, φ〉K := −(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 = Qbu on Γ such that for allv = 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) = 12(∇wu +∇wuT )
Stablizer: s(w, v) =∑
T∈Thh−1T 〈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 = Qbu 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 0h ,
(∇w · uh, q)h − λ−1(ph, q) = 0,∀q ∈ W 0h .
Wh = q : q|T ∈ Pk−1(T ), T ∈ Th W 0h = Wh ∩ L2
0(Ω)
WG-FEM Primal = WG-FEM Mixed
The two weak Galerkin Algorithms are equivalent in the sense thatthe solutions to the two weak Galerkin Algorithms are identical toeach 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 theweak Galerkin finite element solution.
|||Qhu−uh|||+λ−12 ‖Qhp−ph‖+|||Qhp−ph|||0 ≤ Chk(‖u‖k+1+‖p‖k),
where C is a generic constant independent of (u; p). Consequently,
|||u− uh|||+ λ−12 ‖p − ph‖+ |||p − ph|||0 ≤ Chk(‖u‖k+1 + ‖p‖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
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 theweak Galerkin finite element solution. Then, under the regularityassumption, there exists a constant C , such that
‖Q0u− u0‖ ≤ Chk+s(‖u‖k+1 + ‖p‖k
).
Moreover,‖u− u0‖ ≤ Chk+s
(‖u‖k+1 + ‖p‖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
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‖ order ‖ub − Qbu‖ order |||uh − Qhu||| order
2 0.0750 – 0.0424 – 0.3103 –4 0.0192 1.97 0.0115 1.88 0.1566 0.998 0.0049 1.98 0.0031 1.87 0.0787 0.9916 0.0012 1.99 0.0008 1.93 0.0394 1.0032 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.
1h ‖uh − Q0u‖ order ‖ub − Qbu‖ order |||uh − Qhu||| order
2 0.0743 – 0.0424 – 0.3082 –4 0.0190 1.96 0.0113 1.90 0.1555 0.998 0.0048 1.98 0.0031 1.88 0.0782 0.9916 0.0012 1.99 0.0008 1.93 0.0392 1.0032 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
(xy
)
Table: WG based on P1(T )/PRM(e), µ = 0.5, and λ = 1.
1h ‖uh − Q0u‖ order ‖ub − Qbu‖ order |||uh − Qhu||| order
2 0.0352 – 0.0331 – 0.1544 –4 0.0097 1.86 0.0120 1.46 0.0834 0.898 0.0026 1.91 0.0037 1.68 0.0433 0.9416 0.0007 1.96 0.0010 1.87 0.0220 0.9832 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.
1h ‖uh − Q0u‖ order ‖ub − Qbu‖ order |||uh − Qhu||| order
2 0.0344 – 0.0290 – 0.1447 –4 0.0100 1.79 0.0113 1.36 0.0773 0.908 0.0028 1.82 0.0038 1.59 0.0403 0.9416 0.0008 1.90 0.0011 1.81 0.0205 0.9732 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.
1h ‖uh − Q0u‖ order ‖ub − Qbu‖ order |||uh − Qhu||| order
2 0.0341 – 0.0313 – 0.1518 –4 0.0093 1.87 0.0115 1.45 0.0816 0.908 0.0025 1.91 0.0036 1.67 0.0424 0.9516 0.0006 1.96 0.0010 1.86 0.0215 0.9832 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.
1h ‖uh − Q0u‖ order ‖ub − Qbu‖ order |||uh − Qhu||| order
2 0.0340 – 0.0280 – 0.1439 –4 0.0098 1.79 0.0111 1.34 0.0768 0.918 0.0028 1.82 0.0037 1.58 0.0400 0.9416 0.0007 1.90 0.0011 1.81 0.0203 0.9732 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 functionof the velocity of a particle under the influence of drag forcesand 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
∂2ij [Dij f ] = 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
∂2ij(aiju) =g , in Ω,
u =0, on ∂Ω.
assume that a(x) is non-smooth,
weak formulation is given by seeking u such that
d∑i ,j=1
(u, aij∂2ijw) = (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, on Γ ⊂ ∂Ω,∂u
∂n= ψ, on Γ ⊂ ∂Ω.
This is usually an ill-posed problem which does not have a solutionor has many solutions. Let Γc = ∂Ω/Γ. A variational form for thisproblem seeks u ∈ H1
0,Γ(Ω) such that
(∇u,∇w) = (f ,w) + 〈ψ,w〉Γ,
for all w ∈ H10,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 ofthem.
Abstract Problem
Find u ∈ V such that b(u,w) = f (w) for all w ∈ W . Here f is abounded linear functional on W .
Goal: Design finite element methods by using weak Galerkinapproach.
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 ∩ H10
and
b(v ,w) :=d∑
i ,j=1
(v , aij∂2ijw).
For the Cauchy problem for Poisson equation,V ×W = H1
0,Γ(Ω)× H10,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−1T 〈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 by2a11+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 linearfunction).
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 ispiecewise 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 piecewiseconstant).
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 apiecewise 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 aredifferent, 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