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 and upper bounds for the leading 9 exact eigenvalues.

Last updated by Xuefeng LIU, July. 13, 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: Extract the eigenvalue and eigenfunction

Step 5: Draw the eigenfunction

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

Step 6. Lower eigenvalue bounds

Calculate the lower bounds for the leading 9 exact eigenvalues.

Notice

This method only holds for convex domain with homogeneous Dirichlet boundary condition.

Setp 7. 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.