Hybridized discontinuous Galerkin method for elliptic interface problems

TL;DR

This paper introduces a new hybridized discontinuous Galerkin method for elliptic interface problems that achieves optimal error estimates even with limited regularity of the solution, validated by numerical experiments.

Contribution

The paper proposes a novel HDG scheme for elliptic interface problems that does not require flux variables and handles solutions with limited regularity, providing optimal error estimates.

Findings

Achieved optimal convergence rates in HDG and L2 norms.

Validated theoretical results with numerical experiments.

Handled solutions with fractional Sobolev regularity.

Abstract

New hybridized discontinuous Galerkin (HDG) methods for the interface problem for elliptic equations are proposed. Unknown functions of our schemes are $u_h$ in elements and $\hat{u}_h$ on inter-element edges. That is, we formulate our schemes without introducing the flux variable. Our schemes naturally satisfy the Galerkin orthogonality. The solution $u$ of the interface problem under consideration may not have a sufficient regularity, say $u|_{\Omega_1}\in H^2(\Omega_1)$ and $u|_{\Omega_2}\in H^2(\Omega_2)$, where $\Omega_1$ and $\Omega_2$ are subdomains of the whole domain $\Omega$ and $\Gamma=\partial\Omega_1\cap\partial\Omega_2$ implies the interface. We study the convergence, assuming $u|_{\Omega_1}\in H^{1+s}(\Omega_1)$ and $u|_{\Omega_2}\in H^{1+s}(\Omega_2)$ for some $s\in (1/2,1]$, where $H^{1+s}$ denotes the fractional order Sobolev space. Consequently, we succeed in deriving optimal order error estimates in an HDG norm and the $L^2$ norm. Numerical examples to validate our results are also presented.