Abstract/Summary

The solution of partial differential equations modelling water infiltration into soil poses many challenges. The multi-scale and nonlinear nature of soil makes the design of robust and accurate numerical schemes particularly difficult. In addition, error estimation is complicated by low solution regularity. In this thesis, we investigate the mathematical and numerical aspects of the approximation of problems related to subsurface flow by the finite element method. We begin with a variational inequality as a simplified model (albeit of significant interest and complexity in its own right) of a seepage problem. The so-called Signorini problem includes many of the key difficulties, namely nonlinear boundary conditions and lack of dual regularity. We derive rigorous and computable a posteriori error estimates using duality arguments that require careful analysis of primal and dual problems. Crucial in this argument is the design of a novel nonlinear bound-preserving interpolant that respects various inequalities related to the weak form of the problem. These estimates are used to implement a mesh adaptive routine. We then study a physically realistic seepage problem complete with nonlinear coefficients and mixed boundary conditions and inequality constraints. This time, we apply the dual-weighted residual framework of a posteriori error estimation and derive error estimates that are used to optimise the computational mesh for a quantity of interest. The estimates are tested on realistic groundwater scenarios that utilise field data. We conclude with a numerical study of a time-dependent and nonlinear model of two-dimensional subsurface flow. We introduce a method to regularise the nonlinearity in the soil porosity function and derive a posteriori error estimates that account for this approximation in linear elliptic and parabolic cases. We show that in the nonlinear parabolic case, this regularisation mitigates the commonly observed failure of nonlinear solvers for Richards’ equation.