A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems

Abstract

<p style='text-indent:20px;'>We introduce an over-penalized weak Galerkin method for elliptic interface problems with non-homogeneous boundary conditions and discontinuous coefficients, where the method combines a weak Galerkin stabilizer with interior penalty terms. This method employs double-valued functions on interior edges of elements instead of single-valued ones and elements <inline-formula><tex-math id="M1">\begin{document}$ (\mathbb{P}_{k}, \mathbb{P}_{k}, [\mathbb{P}_{k-1}]^{2}) $\end{document}</tex-math></inline-formula>. As an advantage of the method, elliptic interface problems with low regularity are approximated well. The over-penalized weak Galerkin method is based on weak functions whose edge part is double-valued on each interior edge sharing by two neighboring elements. Jumps between the edge parts are naturally used to define penalty terms. The over-penalized weak Galerkin method allows to use arbitrary meshes, even for low regularity solutions. These features make the new method more flexible and efficient for solving interface equations. Furthermore, <i>a priori</i> error estimates in energy and <inline-formula><tex-math id="M2">\begin{document}$ L^{2} $\end{document}</tex-math></inline-formula> norms are derived rigorously, and numerical results confirm the effectiveness of the method.