Lowest-Order Weak Galerkin Finite Element Method for Darcy Flow on Convex Polygonal Meshes

This paper presents the lowest-order weak Galerkin (WG) finite element method for solving the Darcy equation or elliptic boundary value problems on general convex polygonal meshes. In this approach, constants are used in element interiors and on edges to approximate the primal variable (pressure). The discrete weak gradients of these constant basis functions are established in simple $H(div)$-subspaces on polygons that are explicitly constructed by using the normalized coordinates and Wachspress coordinates [W. Chen and Y. Wang, Math. Comp., 86 (2017), pp. 2053--2087]. These discrete weak gradients are used to approximate the classical gradient in the variational formulation. No penalization is needed for this new method. The method results in symmetric positive-definite sparse linear systems. It is locally mass-conservative and produces continuous normal fluxes. The new method has optimal-order convergence in pressure, velocity, and normal flux, when the convex polygon meshes are shape-regular.