High-precision eigenvalue bounds of the Laplacian

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

Question:

Show the high-precision lower and upper bounds for the leading 9 exact eigenvalues.

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: 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.

Lehmann-Goerisch's method: