New class of finite element methods: weak Galerkin methods Xiu Ye - - PowerPoint PPT Presentation

New class of finite element methods: weak Galerkin methods

Xiu Ye

University of Arkansas at Little Rock

Second order elliptic equation

Consider second order elliptic problem:

−∇· a∇u =

f,

in Ω

(1) u

=

0,

• n ∂Ω.

(2) Testing (1) by v ∈ H1

0(Ω) gives

−

• Ω ∇· a∇uvdx =
• Ω

a∇u ·∇vdx −

• ∂Ω

a∇u · nvds =

• Ω

fvdx.

(a∇u, ∇v) = (f, v),

where (f,g) =

• Ω fgdx.

PDE and its weak form

PDE: find u satisfies

−∇· a∇u =

f,

in Ω

u

=

0,

• n ∂Ω.

Its weak form: find u ∈ H1

0(Ω) such that

(a∇u, ∇v) = (f, v), ∀v ∈ H1

0(Ω).

Infinity vs finite

Weak form: find u ∈ H1

0(Ω) such that

(a∇u, ∇v) = (f, v), ∀v ∈ H1

0(Ω).

Let Vh ⊂ H1

0(Ω) be a finite dimensional space.

Continuous finite element method: find uh ∈ Vh such that

(a∇uh,∇vh) = (f,vh), ∀vh ∈ Vh,

Continuous finite element method

Find uh ∈ Vh such that

(a∇uh, ∇vh) = (f, vh), ∀vh ∈ Vh.

Let Vh = Span{φ1,··· ,φn} and uh = ∑n

j=1 cjφj, then n

∑

j=1

(a∇φj,∇φi)cj = (f, φi),

i = 1,··· ,n. The equation above is a symmetric and positive definite linear system. Solve it to obtain the finite element solution uh.

Limitations of the continuous finite element methods

• On approximation functions. Pk only for triangles and Qk for
• quadrilaterals. Hard to construct high order and special elements

such as C1 conforming element.

• On mesh generation. Only triangular or quadrilateral meshes

can be used in 2D. Hybrid meshes or meshes with hanging nodes are not allowed. Not compatible to hp adaptive technique.

Cause and solution

Cause: Continuity requirement of approximating functions cross element boundaries. Solution: Use discontinuous approximations.

Pros and cons of using discontinuous functions

Pros

• Flexibility on approximation functions. Polynomial Pk can be used
• n any polygonal element. Easy to construct high order element.
• Flexibility on mesh generation. Hybrid meshes or meshes with

hanging nodes are allowed. Compatible to hp adaptive technique. Cons

• There are more unknowns.
• Complexity in finite element formulations due to enforcing

connections of numerical solutions between element boundaries.

Weak Galerkin finite element methods

Weak Galerkin (WG) methods use discontinuous approximations. The WG methods keep the advantages:

• Flexible in approximations. Avoid construction of special

elements such as C1 conforming elements.

• Flexible in mesh generation. Hybrid meshes or meshes with

hanging nodes can be used. and minimize the disadvantages:

• Simple formulations.
• Comparable number of unknowns to the continuous finite

element methods if implemented appropriately.

Weak functions

Let T be a quadrilateral with ej for j = 1,··· ,4 as its four sides. Define v =

• v0 ∈ P1(T),

in T 0

vb ∈ P0(e),

• n e ⊂ ∂T

Define Vh(T) = {v ∈ L2(T) : v = {v0,vb}} = span{φ1,··· ,φ7} where

φj =

• 1,
• n ei

0,

• therwise

j = 1,··· ,4

φ5 =

• 1,

in T 0

0,

• n ∂T

φ6 =

• x,

in T 0

0,

• n ∂T

φ7 =

• y,

in T 0

0,

• n ∂T

Weak derivatives

Define a weak gradient ∇wv ∈ [P0(T)]2 for v = {v0,vb} ∈ Vh(T) on the element T:

(∇wv,q)T = −(v0,∇· q)T +vb,q · n∂T , ∀q ∈ [P0(T)]2.

Let φj = {φj,0,φj,b}, j = 1,··· ,7. The definition of the weak gradient gives that for any q ∈ [P0(T)]2

(∇wφ5,q) = −(φ5,0,∇· q)T +φ5,b,q · n∂T = 0.

We have

∇wφ5 = ∇wφ6 = ∇wφ7 = 0.

Using the definition of ∇w, we can find for j = 1,··· ,4

∇wφj = |ej| |T| nj.

Weak gradient ∇w for all the basis function φj can be found explicitly.

The local stiffness matrix for the WG method

Denote Qb the L2 projection to P0(ej). Qbv0|ej = v0(mj) where mj is the midpoint of ej. Define aT(v,w) = (a∇wv,∇ww)T + h−1Qbv0 − vb,Qbw0 − wb∂T. The local stiffness matrix A for the WG method on the element T for second order elliptic problem is a 7× 7 matrix A = (aT(φi,φj)), i,j = 1,··· ,7.

T e1 e2 e3 e4

Weak Galerkin finite element methods

• Define weak function v = {v0,vb} such that

v =

• v0,

in T 0

vb,

• n ∂T

Define weak Galerkin finite element space Vh = {v = {v0,vb} : v0|T ∈ Pj(T),vb ∈ Pℓ(e),e ⊂ ∂T,vb = 0 on ∂Ω}.

• Define a weak gradient ∇wv ∈ [Pr(T)]d for v ∈ Vh on each element T:

(∇wv,q)T = −(v0,∇· q)T +vb,q · n∂T, ∀q ∈ [Pr(T)]d.

Weak Galerkin element: (Pj(T),Pℓ(e),[Pr(T)]d). For example:

(P1(T),P0(e),[P0(T)]d).

Weak Galerkin finite element formulation

Define a(uh,vh) = (a∇wuh,∇wvh)+∑

T

h−1

T u0 − ub,v0 − vb∂T.

The WG method: find uh = {u0,ub} ∈ Vh satisfying a(uh,vh) = (f,vh),

∀vh ∈ Vh.

• Theorem. Let uh be the solution of the WG method associated with local

spaces (Pk(T),Pk(e),[Pk−1(T)]d), then h|||Qhu − uh|||+Qhu − uh ≤ Chk+1uk+1, where Qhu is the L2 projection of u.

Simple formulation: the WG method for the Stokes equations

The weak form of the Stokes equations: find (u,p) ∈ [H1

0(Ω)]d × L2 0(Ω) that

for all (v,q) ∈ [H1

0(Ω)]d × L2 0(Ω)

(∇u,∇v)−(∇· v,p) = (f,v) (∇· u, q) =

0. The 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.

The WG method for the biharmonic equation

The weak form of the Stokes equations: seeking u ∈ H2

0(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.

Implementation of the WG method

The WG method: find uh = {u0,ub} ∈ Vh satisfying a(uh,vh) = (f,vh),

∀vh = {v0,vb} ∈ Vh.

Effective implementation of the WG method:

• 1. Solve u0 as a function of ub from the following local system on element T,

a(uh,vh) = (f,vh),

∀vh = {v0,0} ∈ Vh.

(3)

• 2. Solve ub from the following global system,

a(uh,vh) = (f,vh),

∀vh = {0,vb} ∈ Vh.

(4)

• Theorem. The global system (4) is symmetric and positive definite.

For the lowest order WG method, the number of unknowns of (4) is

# of unknowns = # of interior edges.

The recent development: The WG least-squares method

Consider the model problem,

−∇· a∇u =

f,

in Ω

u

=

0,

• n ∂Ω.

Rewrite the problem as the system of first order equations, q+ a∇u

=

0, in Ω,

∇· q =

f, in Ω, u

=

0,

• n ∂Ω.

The least-squares method: find (q,u) ∈ H(div;Ω)× H1

0(Ω) such that for any

(σ σ σ,v) ∈ H(div;Ω)× H1

0(Ω),

(q+ a∇u,σ σ σ + a∇v)+(∇· q,∇·σ σ σ) = (f,∇·σ σ σ).

The WG Least-squares method

The least-squares method: find (q,u) ∈ H(div;Ω)× H1

0(Ω) such that for any

(σ σ σ,v) ∈ H(div;Ω)× H1

0(Ω),

(q+ a∇u,σ σ σ + a∇v)+(∇· q,∇·σ σ σ) = (f,∇·σ σ σ).

The WG least-squares method: find (qh,uh) ∈ Σh × Vh such that for any

(σ σ σ,v) ∈ Σh × Vh,

(qh + a∇wuh,σ σ σ + a∇wv)+(∇w · qh,∇w ·σ σ σ)+ s1(uh,v)+ s2(qh,σ σ σ) = (f,∇·σ σ σ).

Define

Dh = {ne : ne is unit and normal to e, e ∈ Eh},

Vh = {v = {v0,vb} : v0|T ∈ Pk+1(T),vb|e ∈ Pk(e),e ∈ ∂T,vb = 0, on ∂Ω},

Σh = {σ σ σ = {σ σ σ 0,σ σ σ b} : σ σ σ 0|T ∈ [Pk(T)]d,σ σ σ b|e = σbne,σb|e ∈ Pk(e), e ∈ ∂T}.

Define s1(w,v)

=

∑

T∈Th

h−1Qbw0 − wb, Qbv0 − vb∂T , s2(t,σ

σ σ) =

∑

T∈Th

h(t0 − tb)· n, (σ

σ σ 0 −σ σ σ b)· n∂T ,

The WG least-squares method: find (qh,uh) ∈ Σh × Vh such that for any

(σ σ σ,v) ∈ Σh × Vh, (qh + a∇wuh,σ σ σ + a∇wv)+(∇w · qh,∇w ·σ σ σ)+ s1(uh,v)+ s2(qh,σ σ σ) = (f,∇·σ σ σ).

We introduce a norm |||·|||V in Vh as

|||v|||2

V = ∑ T∈Th

∇wv2

T + s1(v,v),

and a norm |||·|||Σ in Σh as

|||σ σ σ|||2

Σ = ∑ T∈Th

∇w ·σ σ σ2

T +σ

σ σ 02 + s2(σ σ σ,σ σ σ).

• Lemma. There is a constant C such that for all (σ

σ σ,v) ∈ Σh × Vh

C(|||σ

σ σ|||2

Σ +|||v|||2 V) ≤ a(v,σ

σ σ;v,σ σ σ).

• Theorem. Assume the exact solution u ∈ Hk+2(Ω) and q ∈ [Hk+1(Ω)]d.

Then, there exists a constant C such that

|||uh − Qhu|||V +|||qh − Qhq|||Σ ≤ Chk+1(uk+2 +qk+1).

Implementation of the WG least-squares method

The WG least-squares method: find (qh,uh) ∈ Σh × Vh such that for any

(σ σ σ,v) ∈ Σh × Vh,

(qh + a∇wuh,σ σ σ + a∇wv)+(∇w · qh,∇w ·σ σ σ)+ s1(uh,v)+ s2(qh,σ σ σ) = (f,∇·σ σ σ).

Effective implementation of the WG least-squares method:

• 1. Solve the local systems on each element T ∈ Th for any v = {v0, 0} ∈ Vh(T) and

σ σ σ = {σ σ σ 0,0} ∈ Σh(T),

a(uh,qh;v,σ

σ σ) = (f, ∇w ·σ σ σ)T .

• 2. Solve a global system,

a(uh,qh;v,σ

σ σ) = 0, ∀v = {0,vb} ∈ Vh,σ σ σ = {0,qb} ∈ Σh,

(5) For the WG least-squares method with k = 0,

# of unknowns = 2×# of interior edges.

Example: Let Ω = (0,1)×(0,1) and the exact solution is given by u = x(1− x)y(1− y).

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

h L2 error

• rder

L2 error

• rder

3.2327e-01 6.1187e-02 5.4044e-03 1.6178e-01 2.1245e-02 1.5281 1.5135e-03 1.8386 8.0929e-02 8.4355e-03 1.3335 4.0547e-04 1.9015 4.0847e-02 3.8201e-03 1.1586 1.0480e-04 1.9788 2.0610e-02 1.8519e-03 1.0585 2.6515e-05 2.0091 1.0351e-02 9.1819e-04 1.0187 6.6586e-06 2.0064

The weak Galerkin method for the elliptic interface problems

Consider a elliptic interface problem,

−∇· A∇u =

f, in Ω, u

=

0,

• n ∂Ω\Γ,

[[u]]Γ = ψ,

• n Γ,

[[A∇u · n]]Γ = φ,

• n Γ,

Vh = {v = {v0,vb} : v0|T ∈ Pk(T),vb|e ∈ Pk(e),e ∈ ∂T,vb = 0,on,∂Ω} The weak Galerkin method: find uh ∈ Vh such that

(A∇wuh,∇wv)+∑

T

h−1u0 − ub,v0 − vb∂T = (f,v0)

+ψ,A∇wv · nΓ −φ,vbΓ −ψ,v0 − vbΓ,∀v ∈ Vh.

Elliptic interface problems: Example 1

Ω = (0,1)2 with Ω1 = [0.2,0.8]2 and Ω2 = Ω/Ω1.

Then the exact solution: u =

• 5+ 5(x2 + y2),

if (x,y) ∈ Ω1 x2 + y2 + sin(x + y), if (x,y) ∈ Ω2 Permeability: A =

• 1,

if (x,y) ∈ Ω1 2+ sin(x + y), if (x,y) ∈ Ω2

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

Mesh 1 Mesh 2

0.5 1 0.5 1 −2 2 4 6 8 10 12 2 4 6 8 10

Solution on mesh 1 Solution on mesh 2

Elliptic interface problems: Example 2

The exact solution is u(x,y) =

• x − y2 + 10 if (x,y) ∈ Ω1

ex cosπy otherwise

−1 −0.5 0.5 1 −1 −0.5 0.5 1 Ω1 Ω2

Figure : The interface Γ in Example 2.

Mesh max{h} Gradient Solution L2 error

• rder

L2 error

• rder

Level 1 4.7778e-01 1.8483e+00 5.6857e-01 Level 2 2.3889e-01 7.5111e-01 1.2991 1.4087e-01 2.0130 Level 3 1.1944e-01 3.4091e-01 1.1396 3.5107e-02 2.0044 Level 4 5.9720e-02 1.6283e-01 1.0660 8.7630e-03 2.0023 Level 5 2.9860e-02 7.9658e-02 1.0315 2.1891e-03 2.0011

Figure : The WG approximation of Example 2 on mesh level 5. Left: Numerical solution; Right: Exact solution.

Summary

• The weak Galerkin finite element methods represent advanced

methodology for handling discontinuous functions in finite element procedure.

• The weak Galerkin finite element methods have the flexibility of

using discontinuous elements and the simplicity of using continuous elements.