The lowest-order weak Galerkin finite element method for the Darcy equation on quadrilateral and hybrid meshes

This paper presents the lowest-order weak Galerkin finite element method for solving the Darcy equation on quadrilateral and hybrid meshes consisting of quadrilaterals and triangles. In this approach, the pressure is approximated by constants in element interiors and on edges. The discrete weak gradients of these constant basis functions are specified in local Raviart-Thomas spaces RT[0] for quadrilaterals or RT0 for triangles. These discrete weak gradients are used to approximate the classical gradient when solving the Darcy equation. The method produces continuous normal fluxes and is locally massconservative, regardless of mesh quality. It exhibits expected convergence in pressure, velocity, and flux when the quadrilaterals are asymptotically parallelograms. Implementation is straightforward and results in symmetric positive-definite discrete linear systems. Numerical experiments and comparison with other existing methods are also presented.