The Elliptic Equation

4

The Finite Element Method

4-4

3 Discretize the PDE and the boundary conditions to obtain a linear system Ku = F. The unknown vector u contains the values of the approximate solution at the mesh points, the matrix K is assembled from the coefficients c, a, h, and q and the right-hand side F contains, essentially, averages of f around each mesh point and contributions from g. Once the matrices K and F are assembled, you have the entire MATLAB environment at your disposal to solve the linear system and further process the solution.

More elaborate applications make use of the FEM-specific information returned by the different functions of the toolbox. Therefore we quickly summarize the theory and technique of FEM solvers to enable advanced applications to make full use of the computed quantities.

FEM can be summarized in the following sentence: Project the weak form of the differential equation onto a finite-dimensional function space. The rest of this section deals with explaining the above statement.

We start with the weak form of the differential equation. Without restricting the generality, we assume generalized Neumann conditions on the whole boundary, since Dirichlet conditions can be approximated by generalized Neumann conditions. In the simple case of a unit matrix h, setting g = qr and then letting yields the Dirichlet condition because division with a very large q cancels the normal derivative terms. The actual implementation is different, since the above procedure may create conditioning problems. The mixed boundary condition of the system case requires a more complicated treatment, detailed in the section “The Elliptic System.”

Assume that u is a solution of the differential equation. Multiply the equation with an arbitrary test function v and integrate on Ω:

Integrate by parts (i.e., use Green’s formula) to obtain:

The boundary integral can be replaced by the boundary condition:

q→∞

∇ (

– ⋅(c u∇ ) )v+auv dx fv dx

∫

Ω Ω =

∫

c u∇

( )

∫

Ω ^{⋅∇v+auv dx–}

∫

_∂Ω^{(–qu+g)vds =}

∫

_Ω^{fv dx}

The Elliptic Equation

4-5 Replace the original problem with: Find u such that

This equation is called the variational, or weak, form of the differential equation. Obviously, any solution of the differential equation is also a solution of the variational problem. The reverse is true under some restrictions on the domain and on the coefficient functions. The solution of the variational problem is also called the weak solution of the differential equation.

The solution u and the test functions v belong to some function space V. The next step is to choose an N_p-dimensional subspace . Project the weak form of the differential equation onto a finite-dimensional function space simply means requesting u and v to lie in rather than V. The solution of the finite dimensional problem turns out to be the element of that lies closest to the weak solution when measured in the energy norm (see below). Convergence is guaranteed if the space tends to V as . Since the differential operator is linear, we demand that the variational equation is satisfied for N_p test-functions φi that form a basis, i.e.,

Expand u in the same basis of

and obtain the system of equations c u∇

( )

∫

Ω ^{⋅∇v+auv–fv dx–}

∫

_∂Ω^{(–qu+g)vds = 0, ∀v}

V_N

p⊂V V_N

p

V_N

p

V_N

p N_P→∞

∈ V_N

p

c u∇

( )

∫

Ω ^{⋅∇φi+auφi–fφidx–}

∫

_∂Ω^{(–qu+g)φids = 0, i = 1, ,… Np}

V_N

p

u x( ) U_jφ_j( )x

j=1 Np

∑

=

c∇φ_j

( )

∫

Ω ^{⋅∇φi+aφjφidx+}

∫

_∂Ω^qφjφids

 

 

j=1 Np

∑

^Uj

fφ_idx

∫

Ω

= gφ_ids,

∫

∂Ω ⁱ

+ = 1, ,… N_p

4

The Finite Element Method

4-6

Use the following notations:

(Stiffness matrix) (Mass matrix)

and rewrite the system in the form (K + M + Q)U = F + G.

K, M, and Q are N_p-by-N_p matrices, and F and G are N_p-vectors. K, M, and F are produced by assema, while Q, G are produced by assemb. When it is not necessary to distinguish K, M, and Q or F and G, we collapse the notations to KU = F, which form the output of assempde.

When the problem is self-adjoint and elliptic in the usual mathematical sense, the matrix K + M + Q becomes symmetric and positive definite. Many common problems have these characteristics, most notably those that can also be formulated as minimization problems. For the case of a scalar equation, K, M, and Q are obviously symmetric. If c(x) ≥ δ > 0, a(x) ≥ 0 and q(x) ≥ 0 with q(x) >

0 on some part of , then, if U

≠

^0.

U^T(K + M + Q)U is the energy norm. There are many choices of the test-function spaces. The toolbox uses continuous functions that are linear on each triangle of the mesh. Piecewise linearity guarantees that the integrals defining the stiffness matrix K exist. Projection onto is nothing more than linear interpolation, and the evaluation of the solution inside a triangle is done just in terms of the nodal values. If the mesh is uniformly refined,

approximates the set of smooth functions on Ω.

A suitable basis for is the set of “tent” or “hat” functions φ_i. These are linear on each triangle and take the value 0 at all nodes x_j except for x_i. Requesting φ_i(x_i) = 1 yields the very pleasant property

K_{i j,} (c∇φ_j)

∫

Ω ^⋅∇φidx

=

M_{i j,} aφ_jφ_idx

∫

Ω

=

Q_{i j,} qφ_jφ_ids

∫

∂Ω

=

F_i fφ_idx

∫

Ω

=

G_i gφ_ids

∫

∂Ω

=

Ω

∂

U^T(K+M+Q)U c∇u²

∫

Ω ^{+au2 dx+}

∫

_∂Ω^{qu2ds>0, if U≠0.}

=

V_N

p

V_N

p

V_N

p

The Elliptic Equation

4-7 i.e., by solving the FEM system we obtain the nodal values of the approximate solution. Finally note that the basis function φ_i vanishes on all the triangles that do not contain the node x_i. The immediate consequence is that the integrals appearing in K_i,j, M_i,j, Q_i,j, F_i and G_i only need to be computed on the triangles that contain the node x_i. Secondly, it means that K_i,j and M_i,j are zero unless x_i and x_j are vertices of the same triangle and thus K and M are very sparse matrices. Their sparse structure depends on the ordering of the indices of the mesh points.

The integrals in the FEM matrices are computed by adding the contributions from each triangle to the corresponding entries (i.e., only if the corresponding mesh point is a vertex of the triangle). This process is commonly called assembling, hence the name of the function assempde.

The assembling routines scan the triangles of the mesh. For each triangle they compute the so-called local matrices¹ and add their components to the correct positions in the sparse matrices or vectors. The integrals are computed using the mid-point rule. This approximation is optimal since it has the same order of accuracy as the piecewise linear interpolation.

Consider a triangle given by the nodes P₁, P₂, and P₃ as in the following figure.

1. The local 3-by-3 matrices contain the integrals evaluated only on the current triangle.

The coefficients are assumed constant on the triangle and they are evaluated only in the triangle barycenter.

u x( )_i U_jφ_j( )x_i

j=1 Np

∑

^Ui,

= =

The local triangle P1P2P3

P

P

P 1

2

3

P_c P_b

x

y1

1

4

The Finite Element Method

4-8

The simplest computations are for the local mass matrix m:

where P_c is the center of mass of ∆ P₁P₂P₃, i.e.,

The contribution to the right-hand side F is just

For the local stiffness matrix we have to evaluate the gradients of the basis functions that do not vanish on P₁P₂P₃. Since the basis functions are linear on the triangle P₁P₂P₃, the gradients are constants. Denote the basis functions φ₁, φ₂, and φ₃ such that φ_i(P_i) = 1. If P₂ – P₃ = [x₁,y₁]^T then we have that

and after integration (taking c as a constant matrix on the triangle)

If two vertices of the triangle lie on the boundary , they contribute to the line integrals associated to the boundary conditions. If the two boundary points are P₁ and P₂, then we have

and

where P_b is the mid-point of P₁P₂. m_{i j,} a P( )φ_{c i}( )φx _j( )x dx

∇P1P2P3

∫

^{a P( )c area}--- 1^(∆P12^1P2P3)( +δ_{i j,} ),

= =

P_c P₁+P₂+P₃ ---3

=

f_i f P( )_c area(∆P₁P₂P₃) ---3

=

φ₁

∇ 1

2area(∆P₁P₂P₃) --- y₁

x – ₁

=

k_{i j,} 1

4area(∆P₁P₂P₃)

--- y[ _j,–x_j]c P( )_c y₁ –x₁

=

Ω

∂

Q_{i j,} q P( )_b P₁–P₂

--- 16 ( +δ_{i j,} ), i j, 1 2,

= =

G_i g P( )_b P₁–P₂

---2 , i 1 2,

= =

The Elliptic Equation

4-9 For each triangle the vertices P_m of the local triangle correspond to the indices i_m of the mesh points. The contributions of the individual triangle are added to the matrices such that, e.g.,

This is done by the function assempde. The gradients and the areas of the triangles are computed by the function pdetrg.

The Dirichlet boundary conditions are treated in a slightly different manner.

They are eliminated from the linear system by a procedure that yields a symmetric, reduced system. The function assempde can return matrices K, F, B, and ud such that the solution is u = Bv + ud where Kv = F. u is an N_p-vector, and if the rank of the Dirichlet conditions is rD, then v has N_p – rD components.

K_i

m,int K_i

m,in

← +k_{m n,} , m n, = 1 2 3, ,

4

The Finite Element Method

4-10

在文檔中 Partial Differential Equation Toolbox (頁 155-162)