SUMMARYThe aim of the paper is to study the capabilities of the extended finite element method (XFEM) to achieve accurate computations in non-smooth situations such as crack problems. Although the XFEM method ensures a weaker error than classical finite element methods, the rate of convergence is not improved when the mesh parameter h is going to zero because of the presence of a singularity. The difficulty can be overcome by modifying the enrichment of the finite element basis with the asymptotic crack tip displacement solutions as well as with the Heaviside function. Numerical simulations show that the modified XFEM method achieves an optimal rate of convergence (i.e. like in a standard finite element method for a smooth problem).

Abstract. The purpose of this paper is to present a new fictitious domain approach inspired by the extended finite element method introduced by Moës, Dolbow and Belytschko in [18]. An optimal method is obtained thanks to an additional stabilization technique. Some a priori estimates are established and numerical experiments illustrate different aspects of the method. The presentation is made on a simple Poisson problem with mixed Neumann and Dirichlet boundary conditions. The extension to other problems or boundary conditions is quite straightforward.

A general Nitsche method, which encompasses symmetric and non-symmetric variants, is proposed for frictionless unilateral contact problems in elasticity. The optimal convergence of the method is established both for two and three-dimensional problems and Lagrange affine and quadratic finite element methods. Two and three-dimensional numerical experiments illustrate the theory.