A Hybridized Weak Galerkin Finite Element Scheme for the Stokes Equations

TL;DR

This paper introduces a hybridized weak Galerkin finite element method for the Stokes equations, offering flexible polynomial approximations, handling jumps effectively, and providing optimal error estimates validated by numerical tests.

Contribution

The paper presents a novel HWG method with a Lagrange multiplier for the Stokes equations, enhancing flexibility and accuracy over existing WG methods.

Findings

Optimal order error estimates established

Method effectively handles jumps in functions and fluxes

Numerical tests confirm theoretical results

Abstract

In this paper a hybridized weak Galerkin (HWG) finite element method for solving the Stokes equations in the primary velocity-pressure formulation is introduced. The WG method uses weak functions and their weak derivatives which are defined as distributions. Weak functions and weak derivatives can be approximated by piecewise polynomials with various degrees. Different combination of polynomial spaces leads to different WG finite element methods, which makes WG methods highly flexible and efficient in practical computation. A Lagrange multiplier is introduced to provide a numerical approximation for certain derivatives of the exact solution. With this new feature, HWG method can be used to deal with jumps of the functions and their flux easily. Optimal order error estimate are established for the corresponding HWG finite element approximations for both {\color{black}primal variables} and the Lagrange multiplier. A Schur complement formulation of the HWG method is derived for implementation purpose. The validity of the theoretical results is demonstrated in numerical tests.