Efficient Algorithms for Solving Richards Equation: From Linearized Finite Element Method to Deep Learning

The Richards equation is a classical model that describes flow through unsaturated porous media. Efficient numerical methods and relevant theories are limited by its nonlinearity and degeneracy. We introduce two linearized finite element schemes to solve the Richards equation efficiently. In addition to the linearized finite element scheme based on backward Euler format, we also discuss the multi-level linearized finite element scheme based on the Crank–Nicolson format to improve accuracy. Compared to traditional mesh-based methods, such as the finite difference and the finite element methods, deep learning offers a mesh-free approach by taking advantage of automatic differentiation, which can overcome the limitation in dimension and in complexity of boundary shape, allowing for irregularity. We propose an algorithm to obtain the approximate solution of the Richards equation via deep learning. This method is capable of dealing with the high-dimensional Richards equation with complex boundary conditions.