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 (P k , P k , [P k−1 ] 2). 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, a priori error estimates in energy and L 2 norms are derived rigorously, and numerical results confirm the effectiveness of the method.
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 (P k , P k , [P k−1 ] d), or (P k , P k−1 , [P k−1 ] d), with dimensions of space d = 2, 3. The method is absolutely stable with a constant penalty parameter, which is independent of mesh size and shaperegularity. We prove that for quasi-uniform triangulations, condition numbers of the stiffness matrices arising from the OPWG method are O(h −β 0 (d−1)−1), β 0 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 O(h −2). Optimal error estimates in a discrete H 1-norm and L 2-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.
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