Generalized weak Galerkin methods for H(div), H(curl), and H(div, curl)-elliptic problems

Abstract

This work presents a unified framework for generalized weak Galerkin (gWG) methods applied to two- and three-dimensional elliptic problems in the function spaces H(div), H(curl), and H(div, curl). The proposed methodology introduces generalized discrete differential operators, including weakly defined curl and divergence operators, within the weak Galerkin framework. A key feature of this approach is its flexibility in allowing arbitrary combinations of piecewise polynomial approximations in the interior and on the boundaries of each local polytopal element. Optimal order error estimates in energy norms are established for the resulting gWG method. Furthermore, numerical experiments are conducted to validate the theoretical findings and illustrate the accuracy and efficiency of the proposed method.