Nestor Sánchez, Tonatiuh Sanchez-Vizuet, Manuel Solano:
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 Ω by a polygonal subdomain Ωh 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 Ωh and the true domain Ω. 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 Ωh is also provided.