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

On monotone ill-posed problems in Hilbert spaces

The main aim of this paper is to study convergence rates for an operator method of  regularization to solve nonlinear ill-posed problems involving monotone operators in infinite-dimentional Hilbert space without needing closeness conditions. Then these results are presented in form of  combination with finite-dimentional approximations of the space. An iterative method for solving regularized equation is given and  an example in the theory of singular integral equations is considered for illustration