Approximation of the biharmonic problem using P1 finite elements

Abstract : 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.
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Journal of Numerical Mathematics, De Gruyter, 2011, pp.Volume 19, Issue 1, Pages 1-26
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Robert Eymard, Raphaèle Herbin, Mohamed Rhoudaf. Approximation of the biharmonic problem using P1 finite elements. Journal of Numerical Mathematics, De Gruyter, 2011, pp.Volume 19, Issue 1, Pages 1-26. 〈hal-00825673〉

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