A new over-penalized weak galerkin method. Part I: Second-order elliptic problems

In this article, we propose a new over-penalized weak Galerkin (OPWG) method with a stabilizer for second-order elliptic problems. This method employs double-valued functions on interior edges of elements instead of single-valued ones and elements \\begin{document}$ (\\mathbb{P}_{k}, \\mathbb{P}_{k}, [\\mathbb{P}_{k-1}]^{d}) $\\end{document} , or \\begin{document}$ (\\mathbb{P}_{k}, \\mathbb{P}_{k-1}, [\\mathbb{P}_{k-1}]^{d}) $\\end{document} , with dimensions of space \\begin{document}$ d = 2, \\; 3 $\\end{document} . The method is absolutely stable with a constant penalty parameter, which is independent of mesh size and shape-regularity. We prove that for quasi-uniform triangulations, condition numbers of the stiffness matrices arising from the OPWG method are \\begin{document}$ O(h^{-\\beta_{0}(d-1)-1}) $\\end{document} , \\begin{document}$ \\beta_{0} $\\end{document} being the penalty exponent. Therefore we introduce a new mini-block diagonal preconditioner, which is proven to be theoretically and numerically effective in reducing the condition numbers of stiffness matrices to the magnitude of \\begin{document}$ O(h^{-2}) $\\end{document} . Optimal error estimates in a discrete \\begin{document}$ H^1 $\\end{document} -norm and \\begin{document}$ L^2 $\\end{document} -norm are established, from which the optimal penalty exponent can be easily chosen. Several numerical examples are presented to demonstrate flexibility, effectiveness and reliability of the new method.