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The Finite Element Method

Contents:

- Introduction. History. Simple1D Example.

- Weak Derivatives. Sobolev spaces. Weak formulation.

- Finite Element method. Mesh. Stiffness Matrix.

- Example: Simple boundary value problem in2D.

- Summary. Course Contents MAI0088.

Typeset by FoilTEX 2

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The Finite Element Method (FEM)

- A general purpose technique for solving boundary value problems for elliptic

PDEs in complicated domains.

- Mostly developed by engineers 19551965 (Ray W. Clough, John Argyris,. . . )but building on earlier work by Courant, Rayleigh, Ritz, and Galerkin.

- Rigurous mathematical analysis in Strang and Fix, An Analysis of The FiniteElement Method, 1973.


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The Finite Element Method

Example:Find usuch that

u + 2u= 1, x(0, 1),u(0) = 1, and, u(1) = 1. (1)

Solution:Multiply by a test functionand integrate,

10

(u + 2u)vdx = 1

01 vdx.

Integrate by parts

10

(uv +2uv)dx=u(1)v(1)u(0)v(0)+ 1

0(uv+2uv)dx=

10

vdx.

Restrict test functions to V ={vC1(0, 1) such that v(0) = 0}.


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We obtain a new problem: Find usuch that

10

(uv + 2uv)dx=

10

vdx + 1 v(1) for allvV. (2)

Remarks:

- The problems are equivalent in the sense that any solution u C2(0, 1) of

(2) also satisfies (1) except possibly the boundary value u(0) = 1. Also (2) iscalled the weak formof (1).

- We have a(u, v) =< , v > for all vV, where a(, )is a bilinear operatorand< , >is a linear functional.

- The condition u(0) = 1is called anessentialboundary condition and u(1) = 1is a naturalboundary condition.


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Introduce a grid Sh={xi}Ni=0and a space Vh={v(x) piecewise linear on Sh}.

The discetizedproblem is: Find uhVh such that

a(uh, v) =< , v > for all vVh such thatv(0) = 0,uh(0) = 1.

Remarks:

- The solution uhVh is called the finite element solution.

- Theessentialboundary conditionu(0) = 1appear explicitly in the formulationof the problem.

- The naturalboundary condition u(1) = 1 is included in the definition of thefunctional< , >.


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A basisforVh is

x0 x1 x2 xN

1 xN

0 1 2 N1N

i(x) =

1, x=xi,0, x=xj, j =i.

The finite element solution can be written as a linear combination

uh(x) =

Ni=0

cii(x).

Remark:The boundary value uh(0)= c00(0)=1gives us an equation for c0.


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The relation a(uh, v) =< , v >, for v = j, gives us a system of linearequations,

N

i=0

cia(i, j) =< , j >, j = 1, 2 . . . , N .

or

1 0 . . . 0a(0, 1) . . . a(N, 1)

... ...a(0, N) . . . a(N, N)

c0c1

...cN

=

1< , 1>

...< , N >

RemarkThis is a system Kc=f where K is the stiffness matrixandf is the

load vector.


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Elements are easy to compute

a(i, i+1) =

10

(i

i+1+2ii+1)dx=

h0

{1

h

1

h+2(1

x

h)

x

h}dx=

1

h+

h

6.

xi xi+1

i i+1

h

Remark:Ki,j = 0unlessxi andxj are neighbours since othervise ij 0.

So typically the matrix K is sparse.

Exercise: Compute Ki,i = a(i, i) and fi =< , i >. Since i are simplepolynomials its easy to do this numerically.


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Matlab and Comsol example

Find usuch that

u + 2u= 1, x(0, 1),u(0) = 1, and, u(1) = 1.

(3)

Use Comsol Multiphysics to: (module add comsol and comsol matlab onLinux).

Create the computational grid{xi}.

Speficy the equation and boundary conditions.

Assemble the stiffness matrix and load vector.

Solve the problem and visualize the result.


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Weak Derivatives and Sobolev Spaces

Definition:LetuL2()and suppose u L2()satisfies

(v, ) =(u, x), for allC

0 ().

Thenv is theweak derivativeofu.

Example: = (1, 1)andu= 1 |x|.

(u, x) =

Z 11

uxdx=

Z 01

uxdx+

Z 10

uxdx=

[u]01Z 01uxdx+[u]10

Z 10uxdx=

Z 01

(+1)dx

Z 10

(1)dx= (v, )

u(x)

+1

1

v(x)


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Definition:The Sobolev spaceH1() is the set of functions uL2() thathave weak derivatives in L2().

Remarks:

- The set H1()consists of piecewise differentiable and continous functions.

- The space H1() is a Hilbert spacewith scalar product (u, v)1 and norm

u1= (u, u)1/21 .

- Higher order Sobolev spacesHm()can be defined similarily.

Example:Suppose R2 then

(u, v)1=

uvd +

uxvxd +

uyvyd.


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Weak formulationExample:Find uC2()such that

u=f, in,u=g, onD,u

n =h, onN, D

N

Introduce V = {v H1() andv = 0 onD}. Take a test function v V

and write

vf d =

vud =

v ud +

N

u

nvd +

D

u

nvd =

v ud +

N

hvd.


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The weak formulation is: Find u H1()such that

v ud =

f vd +

N

hvd, for allvV

u=g, onD.

(4)

Remarks

- Can introduce a bilinear operatora(u, v)and a linear functional < , v >.

- The next step is to pick a finite dimensional subspace Sh H1() and

discretize the problem (4).


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Example:

N1

N2 N3 N4

N5

N6N7

N8

N9T1

T2

T3

T4

T5

T6T7

T8T9

Data structures needed:

- A2NmatrixNodes, a3MmatrixTriangles, and a2KmatrixEdges.

Here we have Triangles(:,3)= (7 ,8 ,9)T

.

- Is a node Nk subject to a Dirichlet boundary condition? Is an edge Eksubjectto a Neumann boundary condition?


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Linear basis functions

Introduce a spaceVhH1()of piecewise linear functions on a given mesh Th.

For each node Nk we obtain a basis function k. The functions k are uniqelydetermined by k(xk, yk) = 1, k(xj, yj) = 0 for k = j, and k is linear oneach triangle Ti.

Example:

N1

N2 N3 N4

N5

N6N7

N8

N9T1

T2

T3

T4

T5

T6T7

T8T9

The basis function 8 is non-zero on the triangles T3, T4, T5, T6, and T7.


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Assembly of Stiffness matrix and load vector

The discretized problem is: Find uhVh such that

a(uh, j) =Ni=1

cia(i, j) =< , j >, for allj such that j = 0 onD,

and,

uh(xj, yj) =g(xj, yj), for nodes Nj = (xj, yj)T onD.

Remarks:

- Most elements Ki,j are integrals a(j, i). How to compute them?

- The linear system Kc=fis (very) sparse.


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Note that

a(i, j) =

i jd =Tk

Tk

i jd.

and only3basis functions are non-zero on each triangle Tk.

K=sparse(N,N), f=zeros(N,1)

fork= 1 :M

[N1,N2,N3]=Triangles(:,k);Compute9local contributions ( KN1,N1=a

(Tk)(N1, N1),. . . )

Compute3local contributions to < , j >(that isTk

f N1d,. . . ).

end

This is an Element orientedalgorithm. Have to fix rows corresponding to nodessubject to Dirichlet conditions and need a loop over all the Edges subject to aNeumann conditions to get final Kc=f.


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Comsol Multiphysics 2D Example

u= 0

n u= 0

n

u= 0

u= 0

u= 10


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SummaryThe Finite Element Method compared to Finite Differences

- A general purpose technique for solving (elliptic) PDEs in complex domains.

- Relatively easy to make good software that can handle many different types ofequations and domains.

- The linear system Kc= fis sparse but usually not very structured. Have touse standard sparse linear equation solvers. Less efficient preconditioners.

- Assuming the mesh is of good quality error estimates are similar for Finite

Elements and Finite Differences.

- Can use different types of basis functions, e.g.j C1(), without changing

the codes very much.


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MAI0088 Applied Functional Analysis and FEM

Course contents:

mathematical concepts: weak derivative, weak form of a boundary value

problem, and Sobolev spaces.

Use tools from functional analysis to analyze the finite element method:Existance and Uniqueness. Various error estimates.

Implement the core of a realistic finite element solver. Try a commercialsoftware package.

Computational Techniques: Sparse linear equation solvers. Preconditioning.Multigrid solvers.

Course literature:Braess, Finite elements, Cambridge University press.


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21_oktober_2010