Stability and convergence proofs for a discontinuous-Galerkin-based extended finite element method for fracture mechanics

We prove the optimal convergence of a discontinuous-Galerkin-based extended finite element method for two-dimensional linear elastostatic problems over cracked domains. The method, which we proposed earlier [1], has two distinctive traits: a) it enriches the finite element space with the modes I and II singular asymptotic crack tip fields over a neighborhood of the crack tip termed the enrichment region, and b) it allows functions in the finite element space to be discontinuous across the boundary between the enrichment region and the rest of the domain. The treatment for this discontinuity, generally a non-polynomial function, is facilitated by a specially designed discontinuous Galerkin method based on the Bassi–Rebay numerical flux. The stability of the method is contingent upon an inf–sup condition, which we have proved to hold for any quasiuniform mesh family with sufficiently fine meshes. We have also shown the optimal convergence of the displacement and stress fields, and the convergence of the stress intensity factors extracted as the coefficients of the enrichment functions.