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New class of finite element methods: weakGalerkin methods
Lin Mu, Junping Wang and Xiu Ye
University of Arkansas at Little Rock
Lin Mu, Junping Wang and Xiu Ye New class of finite element methods: weak Galerkin methods
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Finite element method for second order elliptic equation
Consider second order elliptic problem:
−∇ · a∇u = f , in Ω (1)
u = 0, on ∂Ω. (2)
Testing (1) by v ∈ H10 (Ω) gives
−∫
Ω∇ · a∇uvdx =
∫Ωa∇u · ∇vdx −
∫∂Ω
a∇u · nvds =
∫Ωfvdx .
(a∇u, ∇v) = (f , v),
where (f , g) =∫
Ω fgdx
Lin Mu, Junping Wang and Xiu Ye New class of finite element methods: weak Galerkin methods
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Continuous Finite element methods
Weak form: find u ∈ H10 (Ω) such that
(a∇u, ∇v) = (f , v), ∀v ∈ H10 (Ω).
Given Th, let Vh ⊂ H10 (Ω) be a finite element space.
Vh = v ∈ H10 (Ω); v |T ∈ Pk(T ), T ∈ Th.
Continuous Finite element method: find uh ∈ Vh such that
(a∇uh,∇vh) = (f , vh), ∀v ∈ Vh,
Lin Mu, Junping Wang and Xiu Ye New class of finite element methods: weak Galerkin methods
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Continuous finite element methods Vh ⊂ H10 (Ω)
Find uh ∈ Vh such that
(a∇uh, ∇vh) = (f , vh), ∀vh ∈ Vh.
Let Vh = Spanφ1, · · · , φn and uh =∑n
j=1 cjφj , then
n∑j=1
(a∇φj ,∇φi )cj = (f , φi ), i = 1, · · · , n.
Simple formulations and many fewer unknowns.
Lin Mu, Junping Wang and Xiu Ye New class of finite element methods: weak Galerkin methods
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Modern techniques in scientific computing
• hp adaptive technique.• Hybrid mesh.The continuous finite element method is not compatible to thesetechniques.
(a) (b)
Lin Mu, Junping Wang and Xiu Ye New class of finite element methods: weak Galerkin methods
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Difficult to construct continuous elements
• C 0 element: high order element.• C 1 element: Argyris element, polynomial with degree 5.
Solution? Discontinuous elements
Lin Mu, Junping Wang and Xiu Ye New class of finite element methods: weak Galerkin methods
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Discontinuous Galerkin finite element methods
Interior penalty finite element method∑T
(a∇uh,∇vh)T −∑e
((a∇uh, [vh])e − σ(a∇vh, [uh])e)
+ α∑e
h−1([uh], [vh])e = (f , vh).
Weakly over penalized interior penalty finite element method∑T
(a∇uh,∇vh)T + α∑e
h−3(Π0[uh], Π0[vh])e = (f , vh).
Lin Mu, Junping Wang and Xiu Ye New class of finite element methods: weak Galerkin methods
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LDG methods
Find qh ∈ Vh, uh ∈Wh such that
(a−1qh, v) + (∇ · v, uh)Th − 〈uh, v · n〉∂Th = 0, ∀v ∈ Vh
(qh,∇w)Th − 〈qh · n,w〉∂Th = (f ,w), ∀w ∈Wh,
where
uh = uh − β · [uh],
qh = qh+ β[qh]− α[uh].
B. Cockburn and C.-W. Shu, The local discontinuous Galerkinmethod for time-dependent convection-diffusion systems, SIAM J.Numer. Anal., 35 (1998), pp. 24402463.
Lin Mu, Junping Wang and Xiu Ye New class of finite element methods: weak Galerkin methods
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Mixed Hybrid method and the HDG method
Mixed hybrid finite element method: find qh ∈ Vh, uh ∈Wh
and uh ∈ Mh such that
(a−1qh, v)− (∇ · v, uh)Th + 〈uh, v · n〉∂Th = 0, ∀v ∈ Vh
(∇ · qh,w)Th = (f ,w), ∀w ∈Wh,
〈µ,qh · n〉∂Th = 0, ∀µ ∈ Mh.
HDG method: find qh ∈ Vh, uh ∈ Wh and uh ∈Mh such that
(a−1qh, v)− (∇ · v, uh)Th + 〈uh, v · n〉∂Th = 0, ∀v ∈ Vh(∇ · qh,w)Th + τ〈uh − uh,w〉∂Th = (f ,w), ∀w ∈ Wh
〈µ,qh · n〉∂Th + τ〈uh − uh, µ〉∂Th = 0, ∀µ ∈Mh.
Lin Mu, Junping Wang and Xiu Ye New class of finite element methods: weak Galerkin methods
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D. Arnold, F. Brezzi, B. Cockburn and D. Marini, Unified analysisof discontinuous Galerkin methods for elliptic problems,SIAM J.Numer. Anal., 39 (2002), 1749-1779.
S. Brenner, L. Owens, and L. Sung, A weakly over-penalizedsymmetric interior penalty method, Ele. Trans. Numer. Anal.,30(2008), 107-127.
B. M. Fraejis de Veubeke, Displacement and equilibrium models inthe finite element method, in Stress Analysis, O. Zienkiewicz andG. Holister, eds., Wiley, New York, 1965.
B. Cockburn, J. Gopalakrishnan, and R. Lazarov, Unifiedhybridization of discontinuous Galerkin, mixed, and conformingGalerkin methods for second order elliptic problems, SIAM J. Nu-mer. Anal. 47 (2009), 1319-136.
Lin Mu, Junping Wang and Xiu Ye New class of finite element methods: weak Galerkin methods
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What is the right finite element formulation whendiscontinuous approximation functions are used
Starting point of the finite element methods: weak form of thePDE:
(a∇u,∇v) = (f , v).
The right finite element formulations should be similar to thecorresponding weak forms of the PDEs.
For discontinuous approximation function v , ∇v is not well defined.
A natural choice of finite element formulation for discontinuouselements should have the form
(a∇wuh,∇wvh) + s(uh, vh) = (f , vh),
where s(uh, vh) is a stabilizer without tuning parameters
Lin Mu, Junping Wang and Xiu Ye New class of finite element methods: weak Galerkin methods
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Weak Galerkin finite element methods
• Define weak function v = v0, vb such that
v =
v0, in T 0
vb, on ∂T
Define weak Galerkin finite element space
Vh = v = v0, vb : v0|T ∈ Pj(T0), vb ∈ P`(e), e ⊂ ∂T , vb = 0 on ∂Ω.
• Define a discrete weak gradient ∇wv ∈ [Pr (T )]d for v ∈ Vh oneach element T :
(∇wv , q)T = −(v0,∇ · q)T + 〈vb, q · n〉∂T , ∀q ∈ [Pr (T )]d .
Lin Mu, Junping Wang and Xiu Ye New class of finite element methods: weak Galerkin methods
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Weak Galerkin finite element methods
Find uh ∈ Vh ⊂ L2(Ω) such that for any vh ∈ Vh
(a∇wuh,∇wvh) +∑T
h−1T 〈u0 − ub, v0 − vb〉∂T = (f , vh).
Theorem. Let uh be the solution of the WG method associatedwith local spaces (Pk(T ),Pk(e), [Pk−1(T )]d),
h|||Qhu − uh|||+ ‖Qhu − uh‖ ≤ Chk+1‖u‖k+1.
Lin Mu, Junping Wang and Xiu Ye New class of finite element methods: weak Galerkin methods
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Deriving new methods under weak Galerkin methodology
v = v0, vb ∈ Pj(T )× P`(e) and ∇wv ∈ [Pr (T )]d .
Examples:
• Choose local functions spaces (Pj(T ),P`(e), [Pr (T )]d).
Let (Pk(T ),Pk+1(e), [Pk+1(T )]d), the WG method:
(a∇wuh,∇wv) = (f , v0).
• Choose vb to be fixed: v = v0, vb = v , v.
(∇wv , q)T = −(v ,∇ · q)T + 〈v, q · n〉∂T .
Lin Mu, Junping Wang and Xiu Ye New class of finite element methods: weak Galerkin methods
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Modified weak Galerkin method: Find uh ∈ Vh such that
(a∇wuh,∇wv) +∑e
h−1e 〈[uh], [v ]〉e = (f , v), ∀v ∈ Vh.
X. Wang, N. Malluwawadu, F. Gao and T. McMillan, A ModifiedWeak Galerkin Finite Element Method, submitted.
∑T
(a∇uh,∇v)T −∑e
((a∇uh, [v ])e − σ(a∇v, [uh])e)
+ α∑e
h−1([uh], [v ])e = (f , v).
Lin Mu, Junping Wang and Xiu Ye New class of finite element methods: weak Galerkin methods
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Simplifying existing methods
The elliptic equation in mixed form:
(a−1q, v)− (∇u, v) = 0
(∇ · q,w) = (f ,w).
LDG method:
(a−1qh, v) + (∇ · v, uh)Th − 〈uh, v · n〉∂Th = 0,
(qh,∇w)Th − 〈qh · n,w〉∂Th = (f ,w),
uh = uh − β · [uh],
qh = qh+ β[qh] − α[uh].
LDG method in term of weak derivatives with β = 0:
(a−1qh, v)− (∇wuh, v)Th = 0,
−(∇w · qh,w)Th − α〈[uh],wn〉∂Th = (f ,w).
Lin Mu, Junping Wang and Xiu Ye New class of finite element methods: weak Galerkin methods
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HDG method:
(a−1qh, v)− (∇ · v, uh)Th + 〈uh, v · n〉∂Th = 〈g , v · n〉∂Ω, ∀v ∈ Vh−(qh,∇w)Th + 〈qh · n,w〉∂Th = (f ,w), ∀w ∈ Wh,
〈µ, qh · n〉∂Th = 0, ∀µ ∈Mh,
qh = qh + τ(uh − uh)n.
HDG method in term of weak derivative:
(a−1qh, v)− (∇wuh, v)Th = 0,
−(qh,∇ww)Th − τ〈uh − uh,w − w〉∂Th = (f ,w).
HDG method and weak Galerkin method are equivalent when a ispiecewise constant. They are not equivalent in general.
Lin Mu, Junping Wang and Xiu Ye New class of finite element methods: weak Galerkin methods
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Schur Complement of the WG formulation
The WG method: find uh = u0, ub ∈ Vh such that
a(uh, v) = (f , v0), ∀ v = v0, vb ∈ Vh
For uh = u0, ub, solve for u0 in term of ub on T
a(uh, v) = (f , v0)T , ∀v = v0, 0 ∈ Vh,
Denote u0 = D(ub, f ). Find ub satisfies
a(D(ub, f ), ub, v) = 0, ∀v = 0, vb ∈ Vh.
The system above: symmetric, positive definite, fewer unknowns.
Lin Mu, Junping Wang and Xiu Ye New class of finite element methods: weak Galerkin methods
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Weak Galerkin formulation for the Stokes equations
Weak form of the Stokes equations: find(u, p) ∈ [H1
0 (Ω)]d × L20(Ω) that for all (v, q) ∈ [H1
0 (Ω)]d × L20(Ω)
(∇u,∇v)− (∇ · v, p) = (f, v)
(∇ · u, q) = 0.
Weak Galerkin method: find (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.
Lin Mu, Junping Wang and Xiu Ye New class of finite element methods: weak Galerkin methods
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Weak Galerkin formulation for the biharmonic equations
The weak form of the Stokes equations: seeking u ∈ H20 (Ω)
satisfying(∆u,∆v) = (f , v), ∀v ∈ H2
0 (Ω),
Weak Galerkin finite element method: seeking uh ∈ Vh satisfying
(∆wuh, ∆wv) + s(uh, v) = (f , v), ∀v ∈ Vh.
Lin Mu, Junping Wang and Xiu Ye New class of finite element methods: weak Galerkin methods
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Summary
• The weak Galerkin finite element methods represent advancedmethodology for handling discontinuous approximation functions.• The weak Galerkin methodology provide a general framework forderiving new methods and simplifying the existing methods.• Simple formulations imply easy analysis and easy applications.
Lin Mu, Junping Wang and Xiu Ye New class of finite element methods: weak Galerkin methods