Approximation of the Ventcel problem, numerical results

TL;DR

This paper discusses the numerical approximation of the Ventcel eigenvalue problem involving surface differential operators, presenting finite element methods and surprising convergence results validated through classical Laplace examples.

Contribution

It introduces a finite element approach for the Ventcel problem and reports unexpected super-convergence and under-convergence phenomena in numerical results.

Findings

Super-convergence for P^1 finite elements

Under-convergence for P^2 and P^3 elements

Validation through classical Laplace problem examples

Abstract

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.