new class of finite element methods: weak galerkin … is the right nite element formulation when...

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

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


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,


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.


Modern techniques in scientific computing

• hp adaptive technique.• Hybrid mesh.The continuous finite element method is not compatible to thesetechniques.

(a) (b)


Difficult to construct continuous elements

• C 0 element: high order element.• C 1 element: Argyris element, polynomial with degree 5.

Solution? Discontinuous elements


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).


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.


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.


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.


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


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 .


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.


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 .


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).


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).


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.


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.


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.


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.


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.


new class of finite element methods: weak galerkin … is the right nite element formulation when...

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