A stabilizer free, pressure robust, and superconvergence weak Galerkin finite element method for the Stokes Equations on polytopal mesh

TL;DR

This paper introduces a new pressure-robust, stabilizer-free weak Galerkin finite element method for the Stokes equations on polytopal meshes, achieving superconvergence and improved accuracy.

Contribution

It develops a novel pressure-robust weak Galerkin method with superconvergence on polytopal meshes, incorporating an $H$(div)-preserving operator to enhance mass conservation and eliminate locking.

Findings

Achieves one order higher convergence rate for velocity and pressure.

Velocity error is independent of pressure, confirming pressure-robustness.

Numerical experiments validate theoretical convergence and robustness.

Abstract

In this paper, we propose a new stabilizer free and pressure robust WG method for the Stokes equations with super-convergence on polytopal mesh in the primary velocity-pressure formulation. Convergence rates with one order higher than the optimal-order for velocity in both energy-norm and the $L^2$-norm and for pressure in $L^2$-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 confirm the pressure-robustness. For Stokes simulation, our proposed scheme only modifies the body source assembling but remains the same stiffness matrix. Four numerical experiments are conducted to validate the convergence results and robustness.