A Stabilizer-Free, Pressure-Robust, and Superconvergence Weak Galerkin Finite Element Method for the Stokes Equations on Polytopal Mesh
Lin Mu, Xiu Ye, Shangyou Zhang
Abstract
In this paper, we propose a new stabilizer-free and pressure-robust weak Galerkin finite element method for the Stokes equations with superconvergence on polytopal mesh in the primary velocity-pressure formulation. Convergence rates with one order higher than the optimal order for velocity in both the energy norm and the $L^2$-norm and for pressure in the $L^2$-norm are proved in our proposed scheme. The $H$(div)-preserving operator has been constructed based on the polygonal mesh for arbitrary polynomial degrees and employed in the body source assembling to break the locking phenomenon induced by poor mass conservation in the classical discretization. Moreover, the velocity error in our proposed scheme is proved to be independent of pressure and thus we confirm the pressure-robustness. For Stokes simulation, our proposed scheme only modifies the body source assembling but keeps the same stiffness matrix. Four numerical experiments are conducted to validate the convergence results and robustness.