A New Weak Galerkin Finite Element Scheme for the Brinkman Model

Abstract

The Brinkman model describes flow of fluid in complex porous media with a high-contrast permeability coefficient such that the flow is dominated by Darcy in some regions and by Stokes in others. A weak Galerkin (WG) finite element method for solving the Brinkman equations in two or three dimensional spaces by using polynomials is developed and analyzed. The WG method is designed by using the generalized functions and their weak derivatives which are defined as generalized distributions. The variational form we considered in this paper is based on two gradient operators which is different from the usual gradient-divergence operators for Brinkman equations. The WG 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 corresponding WG finite element solutions in various norms. Some computational results are presented to demonstrate the robustness, reliability, accuracy, and flexibility of the WG method for the Brinkman equations.

A weak Galerkin finite element scheme for solving the stationary Stokes equations

“…The numerical method of [17] is based on the traditional gradient-divergence variational form for the Brinkman equations. In [24], we presented a new WG scheme based on the gradient-gradient variational form. It is shown that this scheme is suit for the mixed formulation of Darcy which would present a better approximation for this case.…”

Convergence analysis of a modified weak Galerkin finite element method for Signorini and obstacle problems

“…Here we have used the assumption that the set of transition points from u = 0 and u > 0 on C is finite. Combining (28)- (30), and (32), we find that…”

“…The weak Galerkin (WG) method was initially proposed in [19] by Wang and Ye to solve second order elliptic equations. Later on, WG methods have been further developed to solve Helmholtz equations [20], biharmonic equations [21][22][23][24][25], Stokes equations [26,27], Brinkman problems [28][29][30], Maxwell equations [31] and other problems [32][33][34][35]. By using the weak derivative, the WG method allows for totally discontinuous functions of piecewise polynomials on partitions.…”

Numerical Computation of Brinkman Flow with Stable Mixed Element Method

“…Reference [10] gives a stable least-square formulation by introducing an auxiliary variable to substitute for velocity flux. Other more novel methods, such as weak Galerkin finite element method [11,12], multigrid method [13], and virtual method [14], are applied to Brinkman model without considering parameter dependence of solution.…”