Simplified Weak Galerkin and New Finite Difference Schemes for the Stokes Equation

This article presents a simplified formulation for the weak Galerkin finite element method for the Stokes equation without using the degrees of freedom associated with the unknowns in the interior of each element as formulated in the original weak Galerkin algorithm. The simplified formulation preserves the important mass conservation property locally on each element and allows the use of general polygonal partitions. A particular application of the simplified weak Galerkin on rectangular partitions yields a new class of 5- and 7-point finite difference schemes for the Stokes equation. An explicit formula is presented for the computation of the element stiffness matrices on arbitrary polygonal elements. Error estimates of optimal order are established for the simplified weak Galerkin finite element method in the H^1 and L^2 norms. Furthermore, a superconvergence of order O(h^{1.5}) is established on rectangular partitions for the velocity approximation in the H^1 norm at cell centers, and a similar superconvergence is derived for the pressure approximation in the L^2 norm at cell centers. Some numerical results are reported to confirm the convergence and superconvergence theory.Comment: 32 pages, 7 figures, 2 table

A Locking-Free P0P_0 Finite Element Method for Linear Elasticity Equations on Polytopal Partitions

This article presents a $P_0$ finite element method for boundary value problems for linear elasticity equations. The new method makes use of piecewise constant approximating functions on the boundary of each polytopal element, and is devised by simplifying and modifying the weak Galerkin finite element method based on $P_1/P_0$ approximations for the displacement. This new scheme includes a tangential stability term on top of the simplified weak Galerkin to ensure the necessary stability due to the rigid motion. The new method involves a small number of unknowns on each element; it is user-friendly in computer implementation; and the element stiffness matrix can be easily computed for general polytopal elements. The numerical method is of second order accurate, locking-free in the nearly incompressible limit, ease polytopal partitions in practical computation. Error estimates in $H^1$, $L^2$, and some negative norms are established for the corresponding numerical displacement. Numerical results are reported for several 2D and 3D test problems, including the classical benchmark Cook's membrane problem in two dimensions as well as some three dimensional problems involving shear loaded phenomenon. The numerical results show clearly the simplicity, stability, accuracy, and the efficiency of the new method