Approximation of the biharmonic problem using P1 finite elements

We study in this paper a P1 finite element approximation of the solution in H 0 (Ω) of a biharmonic problem. Since the P1 finite element method only leads to an approximate solution in H 0 (Ω), 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 0 (Ω) 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(Ω). 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.