12.1 Introduction and algorithm

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12.1 Introduction and algorithm

In this chapter, we look at eigenvalue problems posed by elliptic PDEs in a domain subject to homogeneous boundary conditions. This class of problem is very large. For example, the natural vibration of structures leads to eigenvalue problems of this type.

The typical example that we solve here is

$\begin{displaymath} \nabla^2\phi+\lambda\phi=0\mbox{ in }\Omega\end{displaymath}$

subject to $\phi = 0$ on the boundary of $\Omega$ . This problem is to find the eigenvalues $\lambda_n$ and the associated eigenfunctions $\phi_n(x,y)$ . This problem describes the natural vibration of a membrane stretched across $\Omega$ .

When converted to a finite element framework, the problem to be solved is

$\begin{displaymath} (A - \lambda B)\mbox{\boldmath{$u$}} = 0\end{displaymath}$

which we will solve under the normalisation condition $\mbox{\boldmath{$u$}}\cdot B\mbox{\boldmath{$u$}}=1$ .These equations are solved as follows. Suppose that we have an estimate $\{\mbox{\boldmath{$u$}}_n,\lambda_n\}$ for the solution, whilst the exact solution is $\{\mbox{\boldmath{$u$}}^{\ast},\lambda^{\ast}\}$ .Write $\mbox{\boldmath{$du$}} = \mbox{\boldmath{$u$}}^{\ast}-\mbox{\boldmath{$u$}}_n$ and $d\lambda = \lambda^{\ast}-\lambda_n$ . Then we have

$\begin{displaymath} \left[A-(\lambda_n+d\lambda)B\right](\mbox{\boldmath{$u$}}_n... ...du$}})\cdot B(\mbox{\boldmath{$u$}}_n+\mbox{\boldmath{$du$}})=1\end{displaymath}$

from which we see that

$\begin{displaymath} (A-\lambda_n B)\mbox{\boldmath{$u$}}_n+(A-\lambda_n B)\mbox{... ...$u$}}_n+2\mbox{\boldmath{$u$}}_n\cdot B\mbox{\boldmath{$du$}}=1\end{displaymath}$

to first order. If we now solve this pair of equations for $d\lambda$ and $\mbox{\boldmath{$du$}}$ and define $\lambda_{n+1}$ and $\mbox{\boldmath{$u$}}_{n+1}$ by $d\lambda = \lambda_{n+1}-\lambda_n$ and $\mbox{\boldmath{$du$}} = \mbox{\boldmath{$u$}}_{n+1}-\mbox{\boldmath{$u$}}_n$ ,we see that

$\begin{displaymath} \lambda_{n+1}=\lambda_n+\frac{\mbox{\boldmath{$u$}}_n\cdot B... ...{\boldmath{$u$}}_n(A-\lambda_n B)^{-1}B\mbox{\boldmath{$u$}}_n}\end{displaymath}$

$\begin{displaymath} \mbox{\boldmath{$u$}}_{n+1}=(A-\lambda B)^{-1} B\mbox{\boldmath{$u$}}_n(\lambda_{n+1}-\lambda_n)\end{displaymath}$

This pair of equations is equivalent to a Newton-Raphson procedure for finding $\{\mbox{\boldmath{$u$}}^{\ast},\lambda^{\ast}\}$ .

In the next section, we will describe the Fasttalk code that will solve the problem. We will then solve three examples and confirm the solutions. Lastly, we will show how to calculate higher eigenvalues.