The Richards equation, a nonlinear elliptic parabolic equation, is widely used to model infiltration in porous media. We develop a finite element method for solving the Richards equation by introducing a new bounded auxiliary variable to eliminate unbounded terms in the weak formulation of the method. This formulation is discretized using a semi-implicit scheme and the resulting nonlinear system is solved using Newton's method. Our approach eliminates the need of regularization techniques and offers advantages in handling both dry and fully saturated zones. In the proposed techniques, a non-overlapping Schwarz domain decomposition method is used for modeling infiltration in layered soils. We apply the proposed method to solve the Richards equation using the Havercamp and van Genuchten models for the capillary pressure. Numerical experiments are performed to validate the proposed approach, including tests such as modeling flows in fibrous sheets where the initial medium is totally dry, two cases with fully saturated and dry regions, and an infiltration problem in layered soils. The numerical results demonstrate the stability and accuracy of the proposed numerical method. The numerical solutions remain positive in the presence of totally dry zones. The numerical investigations clearly demonstrated the capability of the proposed method to effectively predict the dynamics of flows in unsaturated soils.

翻译：Richards方程作为一种非线性椭圆-抛物型方程，广泛应用于多孔介质中的入渗模拟。本文通过引入新的有界辅助变量来消除方法弱形式中的无界项，从而建立了一种求解Richards方程的有限元方法。该公式采用半隐式格式进行离散化，所得非线性系统通过牛顿法求解。本方法无需正则化技术，且在处理干燥区和完全饱和区方面具有优势。在所提出的技术中，采用非重叠Schwarz区域分解法模拟分层土壤中的入渗过程。我们应用所提方法，结合Havercamp和van Genuchten毛细压力模型求解Richards方程。通过数值实验验证了所提方法的有效性，包括模拟初始介质完全干燥的纤维片材流动、完全饱和与干燥区域共存的两种工况，以及分层土壤中的入渗问题。数值结果证明了所提数值方法的稳定性和精确性。在完全干燥区域存在的情况下，数值解仍能保持正值。数值研究清楚地表明，所提方法能有效预测非饱和土壤中流动的动力学行为。