weak galerkin finite element methods for partial...

Weak Galerkin Finite Element Methods for

Partial Differential Equations

Junping WangDivision of Mathematical Sciences

National Science Foundation

Collaborators: Xiu Ye, Lin Mu, Yanqiu Wang, Shan Zhao,Guowei Wei

April 20, 2012

Junping Wang Division of Mathematical Sciences National Science Foundation [21pt] Collaborators: Xiu Ye, Lin Mu, Yanqiu Wang, ShanWeak Galerkin Finite Element Methods for Partial Differential Equations

Talk Outline

Standard Galerkin Finite Elements

Strong and Weak Derivatives

Discrete Weak Derivatives/Differential Operators

Weak Galerkin (WG) Finite Element Methods

Mass Conservation

Approximation and Convergence

WG for the Biharmonic Equation

WG for Stokes Equations

WG for Helmholtz

Numerical Experiments


Weak Form and Galerkin FEMs

The model problem seeks u such that

(a(x)u) = f , in ,

with boundary condition u = 0. A weak form seeks u H10 ()satisfying

(au,v) = (f , v), v H10 ().

Procedures for the standard Galerkin finite element method:

Partition into triangles or tetrahedron.

Construct a subspace, denoted by Sh H10 (), using

piecewise polynomials.

Seek for a finite element solution uh from Sh using the aboveweak formulation.


Some Features in the Galerkin FEM

Find uh Sh such that

(auh,v) = (f , v) v Sh.

Sh is a subspace of the space where the exact solution belongsto

Sh must have good approximation properties

functions in Sh are defined in classical ways

the gradient is computed in the classical sense for any Sh

...


A Formal Change of the Standard Galerkin

Formally, we may replace v by any distribution, and v by anotherdistribution, say wv , and seek for a distribution uh such that

(awuh,wv) = (f , v) v Sh.

Critical Issues:

functions in Sh are allowed to be more general (asdistribution) in what format?

the gradient v is computed weakly, also as distributions how?

convergence and accuracy of the resulting schemes?

robustness?


Weak Functions

Space of weak functions:

W (K ) = {v = {v0, vb} : v0 L2(K ), vb H

12 (K )}.

A weak function on the region K refers to a vector-valuedfunction v = {v0, vb} such that v0 L

2(K ) and

vb H12 (K ).

The first component v0 can be understood as the value of vin the interior of K , and the second component vb is the valueof v on the boundary of K .

vb may not be necessarily related to the trace of v0 on Kshould a trace be defined.


Weak Gradient

The dual of L2(K ) can be identified with itself by using thestandard L2 inner product as the action of linear functionals.

With a similar interpretation, for any v W (K ), the weakgradient of v can be defined as a linear functional wv inthe dual space of H(div,K ) whose action on eachq H(div,K ) is given by

(wv , q) :=

K

v0 qdK +

K

vbq nds,

where n is the outward normal direction to K .

Weak gradients become to be strong gradients if weakfunctions are sufficiently smooth (e.g., as restriction ofsmooth functions).


Discrete Weak Gradients

Weak gradients have to be approximated, which leads to discreteweak gradients: w ,r is given by

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.

Robustness of V (K , r)? How should it be designed?


Weak Finite Element Spaces

Sh(j , ): space of discrete weak functions given as follows

Th: partition of the domain

construct local discrete elements

W (T , j) := {v = {v0, vb} : v0 Pj(T ), vb Pj(T )} .

patch local elements together to get a global space

Sh(j) := {v = {v0, vb} : {v0, vb}|T W (T , j),T Th} .

Weak finite element spaces with homogeneous boundary value:

S0h(j) := {v = {v0, vb} Sh(j), vb|T = 0, T Th} .


Weak Galerkin Finite Element Formulation

For the discrete weak gradient w ,r , we use the local spaceV (K , r) to be specified later on. Shall drop the subscript r fromw ,r .

Find uh = {u0; ub} S0h (j) such that

(awuh, wv) + s(uh, v) = (f , v0), v = {v0; vb} S0h (j),

where wv V (K , r) is given on each element T by

T

wv qdT =

T

v0 qdT +

T

vbq nds,

for all q V (K , r).


Desired Properties for w

There are two properties that are preferred for w :

P1: For any v Sh(j), if wv = 0 on T , then one must havev constant on T . In other words, v0 = vb = constant on T ;

P2: Let u Hm()(m 1) be a smooth function on , and Qhube a certain interpolation/projection of u in the finite elementspace Sh(j). Then, w (Qhu) should be a goodapproximation of u.

Are there any examples that make it work as desired?


Example One: Brezzi-Douglas-Marini Type

The Configuration for each v = {v0, vb} is given by

v0 Pj(T ) and vb Pj+1(e)

V (T , r) = [Pj+1(T )]2.

=

The finite element space Sh(j , j + 1) consists of functionsv = {v0, vb} where v0 is a polynomial of degree no more than j inT , and vb is a polynomial of degree no more than j + 1 on T .The space V (T , r) used to define the discrete weak gradientoperator w is given as vector-valued polynomials of degree nomore than j + 1 on T .


Depiction of Example One

A3

A1 A2

Pj+1(e) Pj+1(e)

Pj+1(e)

Pj(T )

The space [Pj+1(T )]2 is used for the computation of w ,j+1.


Example Two: Raviart-Thomas Type

The Configuration for each v = {v0, vb} is given by

v0 Pj(T ) and vb Pj(e)

V (T , r) = [Pj(T )]2 + Pj (T )x, where x = (x1, x2)

T and Pj(T )is the set of homogeneous polynomials of order j in thevariable x.

The finite element space Sh(j , j) consists of functions v = {v0, vb}where v0 is a polynomial of degree no more than j in T

0, and vb isa polynomial of degree no more than j on T . The space V (T , r)used to define the discrete weak gradient operator w is given asthe Raviart-Thomas of order j on T .


Depiction of Example Two

A3

A1 A2

Pj(e) Pj(e)

Pj(e)

Pj(T )

The space [Pj+1(T )]2 + Pj (T )x is used for the computation of

w ,j .


Pros and Cons on the Two Examples

Pros:

If wv = 0, then v is a constant on each element

The L2 projection operator onto S(j , ) satisfies

w (Qhu) = Rh(u), u H1(T ).

Optimal-order error estimates can be derived

No stabilization s(, ) is necessary

Cons:

The finite element partition is still classical (triangles,tetrahedra, etc)

Limited availability of feasible finite elements on each element

Extension to other problems, such as biharmonic, curl-curl,curl-div, curl4, is less straightforward.


General WG Elements

The general WG element has the following configuration for eachv = {v0, vb}

v0 Pj(T ) and vb Pj(e)

V (T , r) = [Pj1(T )]2

The finite element space Sh(j) consists of functions v = {v0, vb}where v0 is a polynomial of degree no more than j in T , and vb isa polynomial of degree no more than j on T . The space V (T , r)used to define the discrete weak gradient operator w is given bypolynomial space of order j 1 on T .


Depiction of the General WG Element

A3

A1 A2

Pj(e) Pj(e)

Pj(e)

Pj(T )

The space [Pj1(T )]2 is used for the computation of w ,j .


Properties of the General WG Element

If wv = 0, then v MAY NOT BE a constant on eachelement

The L2 projection operator onto S(j , ) satisfies

w (Qhu) = Rh(u), u H1(T ).

A stabilization s(, ) is necessary

The finite element partition is sufficiently general to consist ofpolygons or polyhedra with shape regularity

Standard and easy-to-compute finite elements on each element

Extension to other problems, such as biharmonic, curl-curl,curl-div, curl4, is straightforward.


WG-FEM Preserves Mass Conservation

Consider the following equation

(au) + (bu) + cu = f ,

which arises from many physical problems (e.g., solute transport inporous media).

The conservative strong form:

q + cu = f , q = au + bu.

The conservative integral form:

T

q nds +

T

cudT =

T

fdT .


WG-FEM Preserves Mass Conservation

The weak Galerkin conserves mass with a numerical flux

qh = Rh (awuh + bu0)

in the sense that

T

Rh (awuh + bu0) nds +

T

cu0dT =

T

fdT .

Why is this?

The weak Galerkin formula:

T

awuh wv

T

bu0 wv +

T

cu0v0 =

T

fv0

Choosing v = {v0, vb = 0} so that v0 = 1 on T and v0 = 0elsewhere will do

Choosing different test function checks the continuity of thenumerical flux.


The Biharmonic Equation

The goal is to demonstrate how WG-FEM works for other PDEs.

2u = f in ,

u = g on ,

u

n= on .

The weak formulation seeks u such that

(u,v) = (f , v), v H20 ()

plus the boundary conditions. A WG-FEM would formally look like

(wuh,wv) + s(uh, v) = (f , v), v S0h .

How to construct the weak finite element spaces?

How to define a discrete weak Laplacian w?

What kind of stabilization terms should be added?

Convergence, accuracy, robustness, and others?


Weak Laplacian

First we introduce weak functions on the element T .

A weak function on T refers to a function v = {v0, vb, vn}

such that v0 L2(T ), vb H

12 (T ), and vn n H

12 (T ),

where n is the outward normal direction of T on its boundary.

The first component v0 can be understood as the value of vin the interior of T , and the second and the third componentsvb and vn represent v and v on the boundary of T .

Denote by V(T ) the space of all weak functions on T ; i.e.,

V(T ) = {v = {v0, vb, vn} : v0 L2(T ), vb H

12 (T ),

vn n H

12 (T )}.


Weak Laplacian

Introduce G 2(T ) as a subspace of H2(T ) given by

G 2(T ) = { : H1(T ), L2(T )}.

For any G 2(T ), we have n H12 (T ).

Definition of Weak Laplacian

The dual of L2(T ) can be identified with itself by using thestandard L2 inner product as the action of linear functionals. Witha similar interpretation, for any v V(T ), the weak Laplacian ofv = {v0, vb, vn} is defined as a linear functional wv in the dualspace of G 2(T ) whose action on each G 2(T ) is given by

(wv , )T = (v0, )T + vn n, T vb, nT .

where n is the outward normal direction to T .


Discrete Weak Laplacian

Introduce a discrete weak Laplacian operator by approximating win a polynomial subspace of the dual of G 2(T ).

Definition of Discrete Weak Laplacian

A discrete weak Laplacian operator, w ,k , is defined as the uniquepolynomial w ,kv Pk(T ) that satisfies the following equation

(w ,kv , )T = (v0, )T vb, nT + vn n, T

for all Pk(T ).

Shall drop the subscript k from w ,k .


WG Elements for Biharmonic

The WG element for biharmonic equation has the followingconfiguration for each v = {v0, vb, vn}

v0 Pj+2(T ) and vb Pj+2(e)

vn = vn n with vn Pj+1(e)

wv Pj(T ).

The finite element space Sh(j) consists of functionsv = {v0, vb, vnn} where v0 is a polynomial of degree j + 2 in T , vbis a polynomial of degree j + 2 on T , and vn is a polynomial ofdegree j + 1 on T . The space Pj(T ) is used to define the discreteweak Laplacian.


Depiction of the WG Element for Biharmonic

A3

A1 A2

Pj+2 Pj+1(e)Pj+2 Pj+1(e)

Pj+2 Pj+1(e)

Pj+2(T )

The space Pj (T ) is used for the computation of w ,j .


The Helmholtz Equation

Consider the Helmholtz equation:

u k2u = f , in

u n iku = g , on ,

Continuous Galerkin method: find uh Vh, for all vh Vh

(uh,vh) k2(uh, vh) + ik(uh, vh) = (f , vh) + (g , vh).

Weak Galerkin method: find uh = {v0, vb} Vh, for allvh = {v0, vb} Vh

(duh,dvh) k2(u0, v0) + ik(ub, vb) = (f , v0) + (g , vb).


Stokes Equations

Consider the Stokes equations

u + p = f in ,

u = 0 in ,

u = 0 on .

The usual weak formulation seeks u and p such that

(u,v) (p, v) = (f, v) v [H10 ()]2

( u,w) = 0 w L20()

Key Features:

Both gradient and divergence () operators are involved

w and w should be investigated for their coupling.


Weak Galerkin FEMs

Define for k 1 the following weak finite element spaces

Vh = {v = (v0, vb) : v0|T Pk(T )2, vb Pk(e)

2, vb = 0 on }

and

Wh = {q L20() : q|T Pk1(T ) T Th}.

Define discrete divergence w

(w v, q)T = (v0, q)T + vb, qnT , q Pk1(T ).

Define discrete gradient w

(wv , q)T = (v0, q)T + vb, q nT , q V (k,T ),


WG-FEM for Stokes Equations

Weak Galerkin algorithm: find (uh, ph) Vh Wh such that forall (v, q) Vh Wh

(wuh,wv) (w v, ph) = (f, v)

(w uh, q) = 0.

Theorem Assume that the exact solution u Hk+1()2 H10 ()2

and p Hk() L20(). Then, there exists a constant C such that

w (uh Qhu) + ph 2p Chk(uk+1 + pk).


Numerical Examples

Example 1, solution with corner singularity

Let = (0, 1)2 and the exact solution is

u(x , y) = x(1 x)y(1 y)r2+ ,

where r =

x2 + y2 and (0, 1] is a constant. Clearly, we have

u H10 () H1+() and u / H1+(),

where can be any small positive number. We solve the problemwith = 0.5 and = 0.25.


Table: Example 1. Convergence rate for problems with corner singularity, = 0.5.

h d eh e0 eb duh u u0 u ub u

1/8 1.88e-01 6.40e-03 3.81e-02 2.54e-01 1.49e-02 9.86e-021/16 1.36e-01 2.20e-03 1.92e-02 1.84e-01 7.66e-03 7.37e-021/32 9.74e-02 7.62e-04 9.58e-03 1.32e-01 3.89e-03 5.43e-021/64 6.93e-02 2.65e-04 4.77e-03 9.42e-02 1.96e-03 3.96e-021/128 4.92e-02 9.33e-05 2.38e-03 6.69e-02 9.88e-04 2.85e-02

O(hr )r =

0.4852 1.5251 1.0008 0.4827 0.9805 0.4476

Table: Example 1. Convergence rate for problems with corner singularity, = 0.25.


1/8 4.93e-01 1.69e-02 9.19e-02 6.65e-01 2.56e-02 2.03e-011/16 4.18e-01 7.07e-03 5.50e-02 5.66e-01 1.31e-02 1.78e-011/32 3.53e-01 2.94e-03 3.26e-02 4.79e-01 6.72e-03 1.53e-011/64 2.98e-01 1.22e-03 1.93e-02 4.04e-01 3.42e-03 1.30e-011/128 2.51e-01 5.14e-04 1.15e-02 3.40e-01 1.73e-03 1.10e-01

O(hr )r =

0.2437 1.2613 0.7503 0.2417 0.9717 0.2214


Example 2, anisotropic problems

Consider two dimensional anisotropic problem defined in = (0, 1)2,

(Au) = f ,

where

A =

[k2 00 1

], for k 6= 0.

We set the exact solution to be u(x , y) = sin(2x) sin(2ky).Anisotropic triangular meshes are used for solving this problem.That is, we first divide the domain into kn n sub-rectangles, andthen divide each rectangle into two triangles by connecting adiagonal line. The characteristic mesh size is h = 1/n. We testedtwo cases, k = 3 and k = 9.


Table: Example 2. Convergence rate for the anisotropic problem withk = 3.


1/8 1.48e+0 1.95e-02 1.33e-01 2.70e+0 1.29e-01 3.51e-011/16 7.39e-01 5.11e-03 4.79e-02 1.35e+0 6.53e-02 1.82e-011/32 3.69e-01 1.29e-03 1.70e-02 6.80e-01 3.27e-02 9.14e-021/64 1.84e-01 3.24e-04 6.01e-03 3.40e-01 1.63e-02 4.57e-021/128 9.23e-02 8.12e-05 2.12e-03 1.70e-01 8.18e-03 2.28e-02

O(hr )r =

1.0010 1.9793 1.4946 0.9972 0.9975 0.9885

Table: Example 2. Convergence rate for the anisotropic problem withk = 9.


1/4 7.98e+0 6.80e-02 5.99e-01 1.58e+1 2.52e-01 5.84e-011/8 3.89e+0 2.07e-02 2.14e-01 8.18e+0 1.30e-01 3.52e-011/16 1.91e+0 5.43e-03 7.67e-02 4.12e+0 6.53e-02 1.82e-011/32 9.54e-01 1.37e-03 2.72e-02 2.06e+0 3.27e-02 9.15e-021/64 4.76e-01 3.44e-04 9.63e-03 1.03e+0 1.63e-02 4.57e-02

O(hr )r =

1.0161 1.9160 1.4898 0.9857 0.9883 0.9301


Example 3. Stokes problems Set = [0, 1] [0, 1] with threecircle holes center at (0.5, 0.5), (0.2, 0.8), (0.8, 0.8) and has theradius 0.1. The exact solutions u = [u1, u2] and p are

u1 = x + x2 2xy + x3 3xy2 + x2y

u2 = y 2xy + y2 3x2y + y3 xy2

p = xy + x + y + x3y2 4/3.

The initial mesh is first generated by using MATLAB with defaultsetting, see Fig. . Next, the mesh is refined uniformly for fourtimes. The WG solutions on mesh level 1 and mesh level 5 areshown.


Table: Convergence rate for inhomogeneous Dirichlet boundary valueproblem by P0(T ) P0(T ) RT0(T ) on the triangular mesh.

h d (uh Qhu) uh Qhu ||qh q||

Level 1 1.5123e-001 6.5055e-003 2.2512e-001

Level 2 7.6271e-002 1.7736e-003 9.6785e-002

Level 3 3.8276e-002 4.6167e-004 4.0673e-002

Level 4 1.9168e-002 1.1707e-004 2.3700e-002

Level 5 9.5900e-003 2.9394e-005 1.9385e-002

O(hr), r = 9.9506e-001 1.9501 9.1053e-001


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1 Numerical Solution U1

1

0.5

0

0.5

1

1.5

2

2.5

Figure: Numerical Solution of Level 1 (Left) and Level 5 (Right)


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1 Numerical Solution U2

4

3

2

1

0



0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1 Numerical Solution P

1

0.5

0

0.5

1



Example 4: A non-convex Helmholtz problem

Let f = 0 in (??) and the boundary condition is simply taken as aDirichlet one: u = g on . Here g is prescribed according to theexact solution

u = J(k

x2 + y2) cos( arctan(y/x)). (1)

1 0.5 0 0.5 11

0.5

0

0.5

1


1 0.5 0 0.5 11

0.5

0

0.5

1

0.3

0.2

0.1

0

0.1

0.2

0.3

0.4

0.5

Figure: WG solutions for the non-convex Helmholtz problem with k = 4and = 1. Left: Mesh level 1; Right: Mesh level 6.


1 0.5 0 0.5 11

0.5

0

0.5

1

0.5

0

0.5

Figure: WG solutions for the non-convex Helmholtz problem with k = 4and = 3/2. Left: Mesh level 1; Right: Mesh level 6.


Example 5: Large wave numbers

f = sin(kr)/r in (??), where r =

x2 + y2. The boundary data gin the Robin boundary condition is chosen so that the exactsolution is given by

u =cos(kr)

k

cos k + i sin k

k(J0(k) + iJ1(k))J0(kr) (2)

where J(z) are Bessel functions of the first kind.

1 0.5 0 0.5 11

0.5

0

0.5

1


100

101

102

103

104

103

102

101

100

101

1/h

Re

lativ

e H

1

err

or

k=100kh=0.25

k=50

k=5

k=10

0 50 100 150 200 2500

0.2

0.4

0.6

0.8

1

k

Re

lativ

e H

1

err

or

kh=0.5kh=1kh=0.75

Figure: Relative H1 error of the WG solution. Left: with respect to 1/h;Right: with respect to wave number k .


Figure: Exact solution (left) and piecewise constant WG approximation(right) for k = 100, and h = 1/60.


Figure: Exact solution (left) and piecewise linear WG approximation(right) for k = 100, and h = 1/60.


Where to Find References?

Google Weak Galerkin

Visit arxiv.org, and search for Junping Wang or WeakGalerkin


Thank You for Listening


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