Adaptive regularization for the Richards equation

The Richards equation is ubiquitous in the modelling of flows in porous media. It serves as a model in its own right, but also as a stepping stone to more complex models of multiphase flows. Despite its relative simplicity, it features many challenges from a computational point of view due to the nonsmooth and degenerate nature of the nonlinear state functions. In this paper, we replace these functions with regularized (smooth and nondegenerate) counterparts where the amount of added regularization is controlled by a single regularization parameter. We introduce a set of a simple a posteriori error estimators that we use to adaptively steer the regularization and linearization. In particular, we stop the iterative linearization when the corresponding estimator is dominated by the regularization estimator and adaptively choose the regularization parameter so that the regularization estimator does not dominate the discretization one. The full adaptive algorithm is tested on a suite of numerical examples adapted from recent works on improving the robustness of solvers for the Richards equation and on benchmark cases.