# Solve eigenvalue problem of the Laplacian by finite element method¶

On a square domain $\Omega$, consider the following eigenvalue problem

$$-\Delta u = \lambda u \mbox{ in } \Omega, u=0 \mbox{ on } \partial \Omega \:.$$

The variational formulation for the above eigenvalue problem is to find $u\in H_0^1(\Omega)$ and $\lambda \in R$ such that $$\int_{\Omega} \nabla u \cdot \nabla v dx = \lambda \int_{\Omega} uvdx \mbox{ for all } v \in H_0^1(\Omega) \:.$$

Below, we show how to solve the eigenvalue problem step by step.

Particular, in this speical case, the first 9 eigenvalue are given by $$\left\{\frac{\lambda_i}{\pi^2} \right\}_{i=1}^{9} = \{2,5,5,8,10,10,13,13,18\} \:.$$

### Objective:¶

• Calculate the lower bounds for the leading 9 exact eigenvalues.

• The Crouzeix-Rarviart (CR) finite element method will be used to provide lower eigenvalue bounds; such a method even works for non-convex domain.

• The lower and upper eigenvalue bounds for L-shaped domain are given at the end of this file.

Last updated by Xuefeng LIU, Sep. 4, 2017

## Step 5: Draw the eigenfunction¶

Please choose eig_index to be from 0 to 8 and check the shape of eigenfunction.

## 6. Lower eigenvalue bounds¶

Calculate the lower bounds for the leading 9 exact eigenvalues. To give lower bound, we need the following error estimation for projection ($P_h$) to CR FEM space

$$\| u - P_h u \| \le 0.1893h \| \nabla(u - P_h u) \|$$

where $h$ is the maximum edge length of triangulation of the domain.

### Notice¶

This method works even for non-convex domain.

## Setp 7. Upper and Lower eigenvalue bounds for L-shaped domain¶

We apply the method described above to L-shaped domain. Please make sure the L_uniform.xml is uploaded to the same folder.

First, let check the mesh for L-shaped domain. Use L_uniform.xml as inital mesh. By refine this mesh, we have dense uiform mesh.