Hybridized discontinuous Galerkin method for elliptic interface problems

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.