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

TL;DR

This paper develops and analyzes an HDG method for semi-linear elliptic problems on curved domains, providing optimal error estimates, a non-linear post-processing technique, and an a posteriori error estimator.

Contribution

It introduces an HDG discretization for semi-linear elliptic problems on curved domains, with proven optimal error bounds and a novel non-linear post-processing method.

Findings

Optimal error estimates under mild assumptions

Enhanced convergence through non-linear post-processing

Reliable a posteriori error estimator for boundary data approximation

Abstract

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.