A Lowest-Order Weak Galerkin Finite Element Method for Stokes Flow on Polygonal Meshes

This paper presents a lowest-order weak Galerkin (WG) finite element method for solving the Stokes equations on convex polygonal meshes. Constant vectors are used separately in element interiors and on edges to approximate fluid velocity, whereas constant scalars are used on elements to approximate the pressure. For the constant vector basis functions, their discrete weak gradients are established in a matrix space that is based on the CW0 space (Chen and Wang, 2017), whereas their discrete weak divergences are calculated as elementwise constants. To circumvent the saddle-point problem, a reduced scheme for velocity is established by using three types of basis functions for the discretely divergence-free subspace. A procedure for subsequent pressure recovery is also developed. Error analysis along with numerical experiments on benchmarks are presented to demonstrate accuracy and efficiency of the proposed new method.