Approximation of the biharmonic problem using P1 finite elements

Robert Eymard (1) , Raphaèle Herbin (2) , Mohamed Rhoudaf (3)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
2 LATP - Laboratoire d'Analyse, Topologie, Probabilités
3 Faculté des Sciences et Techniques

We study in this paper a P1 finite element approximation of the solution in $H^2_0(\O)$ of a biharmonic problem. Since the P1 finite element method only leads to an approximate solution in $H^1_0(\O)$, a discrete Laplace operator is used in the numerical scheme. The convergence of the method is shown, for the general case of a solution with $H^2_0(\O)$ regularity, thanks to compactness results and to the use of a particular interpolation of regular functions with compact supports. An error estimate is proved in the case where the solution is in $C^4(\overline{\O})$. The order of this error estimate is equal to $1$ if the solution has a compact support, and only $1/5$ otherwise. Numerical results show that these orders are not sharp in particular situations.

Domaines: Analyse numérique [math.NA]

Robert Eymard, Raphaèle Herbin, Mohamed Rhoudaf. Approximation of the biharmonic problem using P1 finite elements. Journal of Numerical Mathematics, 2011, pp.Volume 19, Issue 1, Pages 1-26. ⟨hal-00825673⟩