A Divergence-Free Pk CDG Finite Element for the Stokes Equations on Triangular and Tetrahedral Meshes

Abstract

In the conforming discontinuous Galerkin method, the standard bilinear form for the conforming finite elements is applied to discontinuous finite elements without adding any inter-element nor penalty form. The Pk (k ≥ 1) discontinuous finite elements and the Pk−1 weak Galerkin finite elements are adopted to approximate the velocity and the pressure respectively, when solving the Stokes equations on triangular or tetrahedral meshes. The discontinuous finite element solutions are divergence-free and surprisingly H-div functions on the whole domain. The optimal order convergence is achieved for both variables and for all k ≥ 1. The theory is verified by numerical examples.