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Bounding eigenvalue for the Laplacian by using the finite element method
Abstract of the project
This project shows how to apply the conforming and non-conforming FEMs to obtain explicit lower and upper bounds for the eigenvalues of the Laplacian.
Code structure
There are two cases of numerical computation. In each case folder, there are pre-prepared meshes.
- Unit square domain. The eigenvalue problem on this domain has a closed-form expression and the estimated eigenvalue bounds are compared with the exact ones directly.
- L-shaped domain. For this domain, there is no closed-form expression for the eigenvalues and there are eigenfunctions missing the $H^2$-regularity. For this case, the conforming FEM and the Crouzeix-Raviart FEM are used to obtain upper and lower bounds, respectively.
How to run the code
Open file EigenvalueBounds.ipynb and run the cells.
Last updated: August 8, 2024
About the directory
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