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 stabilizer for second-order elliptic problems. This employs double-valued functions on interior edges of elements instead single-valued ones and \begin{document}$ (\mathbb{P}_{k}, \mathbb{P}_{k}, [\mathbb{P}_{k-1}]^{d}) $\end{document}, or id="M2">\begin{document}$ \mathbb{P}_{k-1}, dimensions space id="M3">\begin{document}$ d = 2, \; 3 $\end{document}. The is absolutely stable constant penalty parameter, which independent mesh size shape-regularity. We prove that quasi-uniform triangulations, condition numbers the stiffness matrices arising from OPWG are id="M4">\begin{document}$ O(h^{-\beta_{0}(d-1)-1}) id="M5">\begin{document}$ \beta_{0} $\end{document} being exponent. Therefore introduce mini-block diagonal preconditioner, proven to be theoretically numerically effective in reducing magnitude id="M6">\begin{document}$ O(h^{-2}) Optimal error estimates discrete id="M7">\begin{document}$ H^1 $\end{document}-norm id="M8">\begin{document}$ L^2 established, optimal exponent can easily chosen. Several numerical examples presented demonstrate flexibility, effectiveness reliability method.