Optimal L2 error bounds for finite element approximations of Laplace eigenfuncion

Abstract of the project

For conforming approximations of eigenfunctions of the Laplace operator, we propose a fully computable upper bound on the $L^2$ norm of their error. The bound is based on the explicitly known value of the a priori error estimation for the boundary value problems. The resulting a posteriori error estimate bounds the distance of spaces of exact and approximate eigenfunctions and, hence, is robust even in the case of multiple and tightly clustered eigenvalues.

Code structure

There are two cases of numerical computation. In each case folder, there is the setting.ini file to configure the mesh size for FEM computation.

• Unit square domain. The eigenvalue problem has a closed form and the estimated error bound is compared with the error itsself.
• L-shaped domain. This code displays the etimated bound in case the solution does not have $H^2$-regularity.

How to run the code

Evaluation of the projection error constant:

Enter the folder of Calculate_Ch_by_hypercircle and run the code below in a terminal

python evaluate_projection_constant.py [Case Folder]

Record the computed value of the projectio error constants and set the value in the ipynb files.

In each case folder, open the ipynb files and run the inline code in Jupyterlab.

• Unit square case: Case_UnitSquare/UnitSquare_EigenVectorEst.ipynb
• L-shaped domain: Case_LShape/LShape_EigenVectorEst.ipynb