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

Generalized weak Galerkin finite element methods for second order elliptic problems