A linear domain decomposition method for partially saturated flow in porous media

Abstract

The Richards equation is a nonlinear parabolic equation that is commonly used for modelling saturated/unsaturated flow in porous media. We assume that the medium occupies a bounded Lipschitz domain partitioned into two disjoint subdomains separated by a fixed interface Γ. This leads to two problems defined on the subdomains which are coupled through conditions expressing flux and pressure continuity at Γ. After an Euler implicit discretisation of the resulting nonlinear subproblems, a linear iterative (L-type) domain decomposition scheme is proposed. The convergence of the scheme is proved rigorously. In the last part we present numerical results that are in line with the theoretical finding, in particular the convergence of the scheme under mild restrictions on the time step size. We further compare the scheme to other approaches not making use of a domain decomposition. Namely, we compare to a Newton and a Picard scheme. We show that the proposed scheme is more stable than the Newton scheme while remaining comparable in computational time, even if no parallelisation is being adopted. After presenting a parametric study that can be used to optimise the proposed scheme, we briefly discuss the effect of parallelisation and give an example of a four-domain implementation.