Approximation of the biharmonic problem using P1 finite elements

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