A penalized weak Galerkin spectral element method for second order elliptic equations

Abstract

A weak Galerkin spectral element method is proposed to solve second order partial differential equations. Following the idea of the weak Galerkin finite element method, this method introduces for the unknown solution a weak function consisting of one component on the elements together with one component on the element interfaces, and replaces derivatives in the standard variational form with weak derivatives defined on the space of weak functions on each element. As in a classic spectral element method, approximation spaces for weak functions on each triangular or parallelogram element are defined from orthogonal polynomials on the reference domain through a one-to-one mapping, and approximation spaces for weak derivatives are then established via the Piola transform after an insightful investigation of the weak gradient and the discrete weak gradient. To eliminate the effect of the possible nullity of the discrete weak gradient and guarantee the wellposedness of the resulting algebraic system, a penalty term defined on the edges is supplemented into the Galerkin approximation scheme. Error estimates for both the source problem and the eigenvalue problem on meshes consisting of affine families of triangles and quadrilaterals are obtained in the sequel, which are optimal in the mesh size and suboptimal by one-half order with respect to the polynomial degree. Numerical experiments for the eigenvalue problems are performed on both the typical square domain and L-shaped domain with triangular meshes and quadrilateral meshes, which illustrate the effectiveness and high accuracy of our penalized weak Galerkin spectral element method.