Approximation of the Ventcel problem, numerical results
This is an incremental report on numerical results for a known problem, primarily of interest to researchers in finite element methods for eigenvalue problems with surface operators.
The paper reports numerical approximations of the Ventcel eigenvalue problem, finding unexpected convergence behavior: super-convergence for P1 elements but under-convergence for P2 and P3 elements.
Report on the numerical approximation of the Ventcel problem. The Ventcel problem is a 3D eigenvalue problem involving a surface differential operator on the domain boundary: the Laplace Beltrami operator. We present in the first section the problem statement together with its finite element approximation, the code machinery used for its resolution is also presented here. The last section presents the obtained numerical results. These results are quite unexpected for us. Either super-converging for $P^1$ Lagrange finite elements or under converging for $P^2$ and $P^3$. The remaining sections 2 and 3 provide numerical results either for the classical Laplace or for the Laplace Beltrami operator numerical approximation. These examples being aimed to validate the code implementation.