Convergence analysis of linear or quadratic X-FEM for curved free boundaries

The aim of this paper is to provide a-priori error estimates for problems involving curved interfaces and solved with the linear or quadratic extended finite-element method (X-FEM), with particular emphasis on the influence of the geometry representation and the quadrature. We focus on strong discontinuity problems, which covers the case of holes in a material or cracks not subjected to contact as the main applications. The well-known approximation of the curved geometry based on the interpolated level-set function and straight linear or curved quadratic subcells is used, whose accuracy is quantified by means of an appropriate error measure. A priori error estimates are then derived, which depend upon the interpolation order of the displacement, and foremost upon the above error measure and the quadrature scheme in the subcells. The theoretical predictions are successfully compared with numerical experiments.