Error analysis of an unfitted HDG method for a class of non-linear elliptic problems

TL;DR

This paper analyzes an unfitted HDG method for nonlinear elliptic problems in curved domains, providing error estimates and demonstrating optimal convergence under mild assumptions.

Contribution

It introduces an unfitted HDG discretization for nonlinear elliptic problems on curved domains and establishes well-posedness and optimal error estimates.

Findings

Discrete systems are well posed under mild assumptions.

Error estimates show optimal convergence order.

Method works with polygonal subdomains not interpolating the true boundary.

Abstract

We study Hibridizable Discontinuous Galerkin (HDG) discretizations for a class of non-linear interior elliptic boundary value problems posed in curved domains where both the source term and the diffusion coefficient are non-linear. We consider the cases where the non-linear diffusion coefficient depends on the solution and on the gradient of the solution. To sidestep the need for curved elements, the discrete solution is computed on a polygonal subdomain that is not assumed to interpolate the true boundary, giving rise to an unfitted computational mesh. We show that, under mild assumptions on the source term and the computational domain, the discrete systems are well posed. Furthermore, we provide a priori error estimates showing that the discrete solution will have optimal order of convergence as long as the distance between the curved boundary and the computational boundary remains of the same order of magnitude as the mesh parameter.