Hybridized discontinuous Galerkin method for elliptic interface problems
This work provides a rigorous convergence analysis for HDG methods on elliptic interface problems with reduced regularity, benefiting computational scientists solving interface problems.
The authors propose new hybridized discontinuous Galerkin methods for elliptic interface problems that do not require a flux variable, and derive optimal-order error estimates under low regularity assumptions (u in H^{1+s} with s>1/2). Numerical examples validate the theoretical results.
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 $u|_{Ω_1}\in H^2(Ω_1)$ and $u|_{Ω_2}\in H^2(Ω_2)$, where $Ω_1$ and $Ω_2$ are subdomains of the whole domain $Ω$ and $Γ=\partialΩ_1\cap\partialΩ_2$ implies the interface. We study the convergence, assuming $u|_{Ω_1}\in H^{1+s}(Ω_1)$ and $u|_{Ω_2}\in H^{1+s}(Ω_2)$ 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.