The Modified Weak Galerkin Finite Element Method for Solving Brinkman Equations

Abstract:

      A modified weak Galerkin (MWG) finite element method is introduced for the Brinkman equations in this paper. We approximate the model by the variational formulation based on two discrete weak gradient operators. In the MWG finite element method, discontinuous piecewise polynomials of degree $k$ and $k-1$ are used to approximate the velocity $\textbf{\textit{u}}$ and the pressure $p$, respectively. The main idea of the MWG finite element method is to replace the boundary functions by the average of the interior functions. Therefore, the MWG finite element method has fewer degrees of freedom than the WG finite element method without loss of accuracy. The MWG finite element method satisfies the stability conditions for any polynomial with degree no more than $k-1$. The MWG finite element method is highly flexible by allowing the use of discontinuous functions on arbitrary polygons or polyhedra with certain shape regularity. Optimal order error estimates are established for the velocity and pressure approximations in $H^1$ and $L^2$ norms. Some numerical examples are presented to demonstrate the accuracy, convergence and stability of the method.