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

Abstract

Abstract. A weak Galerkin (WG) finite element method for solving the stationary Stokes equations in two-or three-dimensional spaces by using discontinuous piecewise polynomials is developed and analyzed. The variational form we considered is based on two gradient operators which is different from the usual gradient-divergence operators. 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.Numerical results are presented to illustrate the theoretical analysis of the new WG finite element scheme for Stokes problems.

Simplified weak Galerkin and new finite difference schemes for the Stokes equation

“…8.4. The shape of streamlines is similar to the result given in [10,12,17,21], and the results look quite reasonable.…”

“…The right-hand side function and the Dirichlet boundary data are chosen to match the exact solution. Note that this example has been considered in [21]. The numerical approximation pressure field is shown in Fig.…”

“…Once again, the numerical results outperforms the theoretical prediction of r = 1.5 shown in Theorem 7.1. [10,12,17,21]. In this test case, a uniform mesh with meshsize h = 1/32 is employed in our new finite difference scheme.…”

“…WG-FEM has the flexibility of dealing with discontinuous elements while sharing the same weak formulation with the classical conforming finite element methods. Following its first development in [18,19] for second order elliptic equations, the method has gained a lot attention and popularity and has been applied to various PDEs including the Stokes equation, biharmonic equation, elasticity equation, and Maxwell's equations, see [16,21,11] and the references therein for more details.…”

A note on the optimal degree of the weak gradient of the stabilizer free weak Galerkin finite element method

“…In this paper, we are concerned with the development of an SFWG finite element method using the following Poisson equation The standard weak Glaerkin (WG) method for the problem (1.1)-(1.2) seeks weak Galerkin finite element solution u h = {u 0 , u b } such that (∇ w u h , ∇ w v) + s(u h , v) = (f, v), (1.4) for all v = {v 0 , v b } with v b = 0 on ∂Ω, where ∇ w is the weak gradient operator and s(u h , v) in (1.4) is a stabilizer term that ensures a sufficient weak continuity for the numerical approximation. The WG method has been developed and applied to different types of problems, including convection-diffusion equations [7,6], Helmholtz equations [9,12,5], Stokes flow [11,10], and biharmonic problems [8]. Recently, Al-Taweel and Wang in [2], proposed the lowest-order weak Galerkin finite element method for solving reaction-diffusion equations with singular perturbations in 2D.…”