Weak Galerkin finite element method for second order problems on curvilinear polytopal meshes with Lipschitz continuous edges or faces

In this paper, we propose new basis functions defined on curved sides or faces of curvilinear elements (polygons or polyhedrons with curved sides or faces) for the weak Galerkin finite element method. Those basis functions are constructed by collecting linearly independent traces of polynomials on the curved sides/faces. We then analyze the modified weak Galerkin method for the elliptic equation and the interface problem on curvilinear polytopal meshes with Lipschitz continuous edges or faces. The method is designed to deal with less smooth complex boundaries or interfaces. Optimal convergence rates for H1 and L2 errors are obtained, and arbitrary high orders can be achieved for sufficiently smooth solutions. The numerical algorithm is discussed and tests are provided to verify theoretical findings. •A high-order weak Galerkin method is proposed for second order PDEs with Lipschitz continuous curved boundaries/interfaces.•The definition of basis functions on curved sides/faces is presented. These functions are constructed by collecting linearly independent traces of polynomials.•The proposed method can attain arbitrarily high orders for sufficiently smooth solutions. The rate of convergence is independent of the geometry.•Implementation of the algorithm is discussed. Tests are presented to verify our theoretical findings.