An Upwind Weak Galerkin Finite Element Method for the Two‐Dimensional Unsaturated Soil Water Flow Problem

ABSTRACT In this paper, a novel upwind weak Galerkin (WG) finite element method is proposed to solve the two‐dimensional unsaturated soil water flow problem. The proposed numerical scheme combines the classical weak Galerkin finite element method with an upwind stabilization technique to enhance the stability of the numerical solution. The well‐posedness of the numerical scheme is established, and optimal error estimates are derived in the energy norm and the norm. Numerical experiments are conducted to verify the theoretical analysis and demonstrate the efficiency of the proposed method.