A priori and a posteriori error analysis of an unfitted HDG method for semi-linear elliptic problems

We present a priori and a posteriori error analysis of a high order hybridizable discontinuous Galerkin (HDG) method applied to a semi-linear elliptic problem posed on a piecewise curved, non polygonal domain. We approximate \(\Omega\) by a polygonal subdomain \(\Omega_h\) and propose an HDG discretization, which is shown to be optimal under mild assumptions related to the non-linear source term and the distance between the boundaries of the polygonal subdomain \(\Omega_h\) and the true domain \(\Omega\). Moreover, a local non-linear post-processing of the scalar unknown is proposed and shown to provide an additional order of convergence. A reliable and locally efficient a posteriori error estimator that takes into account the error in the approximation of the boundary data of \(\Omega_h\) is also provided.