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.

提出了一种弱Galerkin谱元方法来求解二阶偏微分方程。遵循弱Galerkin有限元方法的思想，此方法为未知解引入了一个弱函数，该函数由元素上的一个分量和元素接口上的一个分量组成，并用定义为的微分导数替换标准变分形式的导数。每个元素上弱函数的空间。与经典的光谱元素方法一样，每个三角形或平行四边形元素上的弱函数的近似空间是通过参考域上的正交多项式通过一对一映射定义的，然后在对弱梯度和离散弱梯度进行深入研究之后，通过Piola变换建立弱导数的近似空间。为了消除离散弱梯度可能为零的影响并保证所得代数系统的适定性，在Galerkin近似方案中补充了在边沿上定义的惩罚项。在续集中获得了由三角形和四边形的仿射族组成的网格的源问题和特征值问题的误差估计，它们在网格大小上是最优的，并且相对于多项式次优为二分之一。在具有三角形网格和四边形网格的典型正方形域和L形域上都进行了特征值问题的数值实验，