A Hybridized Weak Galerkin Finite Element Scheme for the Stokes Equations
For computational scientists solving Stokes flow problems, this method offers a flexible and efficient approach that handles function and flux jumps, though it is an incremental extension of existing weak Galerkin methods.
This paper introduces a hybridized weak Galerkin finite element method for solving Stokes equations, achieving optimal order error estimates for both primal variables and the Lagrange multiplier, as validated by numerical tests.
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.