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 $$ \left\{\frac{\lambda_i}{\pi^2} \right\}_{i=1}^{9} = \{2,5,5,8,10,10,13,13,18\} \:. $$


Last updated by Xuefeng LIU, Sep. 4, 2017

Step 1 : Mesh generation and FEM space definition

Step 2: Variational formulation

Step 3: Calculate matrix and solve the matrix eigenvalue problem

Step 4: Approximate eigenvalues obtained by CR FEM

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.


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.

7.1 Lower bound

7.2 Upper bound

7.3 Draw the lower bound and upper bound in graph