This article proposes and analyzes the generalized weak Galerkin ({\rm g}WG) finite element method for the second order elliptic problem. A generalized discrete weak gradient operator is introduced in the weak Galerkin framework so that the {\rm g}WG methods would not only allow arbitrary combinations of piecewise polynomials defined in the interior and on the boundary of each local finite element, but also work on general polytopal partitions. Error estimates are established for the corresponding numerical functions in the energy norm and the usual $L^2$ norm. A series of numerical experiments are presented to demonstrate the performance of the newly proposed {\rm g}WG method.

翻译：本文针对二阶椭圆问题提出并分析了广义弱Galerkin ({\rm g}WG) 有限元方法。在弱Galerkin框架中引入广义离散弱梯度算子，使得{\rm g}WG方法不仅允许在每个局部有限元内部和边界上定义的分片多项式任意组合，而且适用于一般多边形剖分。建立了数值函数在能量范数和通常$L^2$范数下的误差估计。通过一系列数值实验验证了新提出的{\rm g}WG方法的性能。