Gauss–Newton Multilevel Methods for Least-Squares Finite Element Computations of Variably Saturated Subsurface Flow

Abstract

We apply the least-squares mixed finite element framework to the nonlinear elliptic problems arising in each time-step of an implicit Euler discretization for variably saturated flow. This approach allows the combination of standard piecewise linear H¹-conforming finite elements for the hydraulic potential with the H(div)-conforming Raviart–Thomas spaces for the flux. It also provides an a posteriori error estimator which may be used in an adaptive mesh refinement strategy. The resulting nonlinear algebraic least-squares problems are solved by an inexact Gauss–Newton method using a stopping criterion for the inner iteration which is based on the change of the linearized least-squares functional relative to the nonlinear least-squares functional. The inner iteration is carried out using an adaptive multilevel method with a block Gauss–Seidel smoothing iteration. For a realistic water table recharge problem, the results of computational experiments are presented.