Fast-secant algorithms for the non-linear Richards' equation

Groundwater flow in partially saturated porous media is modelled by using the non-linear Richards equation, which is discretized in the present work by using linear mixed-hybrid finite elements. The discretization produces an algebraic non-linear system, which can be solved by an iterative fixed-point algorithm, the Picard method. The convergence rate is linear, and may be top poor for practical applications. A superlinear convergence rate is obtained by considering a Broyden-type approach, based on the Shermann-Morrison formula. The local character of the Broyden method can be overcome by an accurate estimate of the initial solution, that is by appropriately initializing the computation via some (relaxed) Picard iterations. This strategy needs a convergence criterion to decide when switching from the Picard to the quasi-Newton method, which is crucial for the effectiveness of the scheme, as illustrated by some numerical experiments. We also consider the non-linear algebraic problem from a different viewpoint. Instead of applying the quasi-Newton method directly to such a non-linear system, we applied it to the non-linear function tied to the Picard scheme. Each function evaluation requested by such an algorithm corresponds to a local step of the Picard method, which is then used to compute a Broyden displacement. The present technique can be seen as an accelerated Picard algorithm. We compare the performances of these algorithms when applied to a stationary and a time-dependent benchmark problem.