A study on iterative methods for solving Richards` equation

This work concerns linearization methods for efficiently solving the Richards` equation,a degenerate elliptic-parabolic equation which models flow in saturated/unsaturated porous media.The discretization of Richards` equation is based on backward Euler in time and Galerkin finite el-ements in space. The most valuable linearization schemes for Richards` equation, i.e. the Newtonmethod, the Picard method, the Picard/Newton method and theLscheme are presented and theirperformance is comparatively studied. The convergence, the computational time and the conditionnumbers for the underlying linear systems are recorded. The convergence of theLscheme is theo-retically proved and the convergence of the other methods is discussed. A new scheme is proposed,theLscheme/Newton method which is more robust and quadratically convergent. The linearizationmethods are tested on illustrative numerical examples

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

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 $\Gamma$. This leads to two problems defined on the subdomains which are coupled through conditions expressing flux and pressure continuity at $\Gamma$. 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 unconditional convergence of the scheme. 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. Finally we present a parametric study that can be used to optimize the proposed scheme.Comment: 34 pages, 13 figures, 7 table

Unsaturated subsurface flow with surface water and nonlinear in- and outflow conditions

We analytically and numerically analyze groundwater flow in a homogeneous soil described by the Richards equation, coupled to surface water represented by a set of ordinary differential equations (ODE's) on parts of the domain boundary, and with nonlinear outflow conditions of Signorini's type. The coupling of the partial differential equation (PDE) and the ODE's is given by nonlinear Robin boundary conditions. This article provides two major new contributions regarding these infiltration conditions. First, an existence result for the continuous coupled problem is established with the help of a regularization technique. Second, we analyze and validate a solver-friendly discretization of the coupled problem based on an implicit-explicit time discretization and on finite elements in space. The discretized PDE leads to convex spatial minimization problems which can be solved efficiently by monotone multigrid. Numerical experiments are provided using the DUNE numerics framework.Comment: 34 pages, 5 figure

Mixed finite element approximations for Darcy flow of isentropic gases

In this paper, the mixed finite element methods are analyzed for the approximation of the solution of the system of equations that describes the single-phase Darcy flow of isentropic gas in a porous medium. Our numerical approach is based on the mixed finite element method (MFEM) in space, and backward-differences in time. The lowest order Raviart-Thomas elements are used. Within this frame work, we derive error estimates in suitable norms and show the convergence of the scheme. The features of the MFEM, especially of the lowest order Raviart- Thomas elements, are now fully exploited in the proof of convergence. Finally, we give the numerical experiments to confirm the theoretical analysis regarding convergence rates