Least-Squares Mixed Finite Elements for a Nonlinear Elliptic Problem: A Gauss-Newton Multilevel Method

Gerhard Starke
Universität-GH Essen, Germany

Abstract

Least-squares finite element methods for first-order system formulations of linear second-order elliptic boundary value problems have received much attention in recent years. In this talk, this methodology is applied to a nonlinear problem arising from an implicit time discretization of variably saturated subsurface flow. This approach simultaneously constructs approximations to the flux in Raviart-Thomas spaces and to the hydraulic potential by standard H1-conforming linear finite elements. Two important properties of the least-squares approach which will be utilized are:

The first of these properties leads to adaptive refinement strategies based on the local least-squares functional which allow the resolution of the steep saturation fronts which occur as water infiltrates dry soil. The focus of this presentation is on the second property, in particular, on an inexact version of the Gauß-Newton method which combines robustness with efficiency.

Adaptive multilevel methods are used for the linear least-squares problems arising in each step of the Gauß-Newton method. For the Raviart-Thomas spaces this requires an adaptation of the multilevel method by Arnold, Falk and Winther to locally refined triangulations. The crucial part of the Gauss-Newton multilevel method is the proper choice of the stopping criterion for the inner (multilevel) iteration based on the error of the outer (Gauss-Newton) iteration. In the context of nonlinear least-squares finite element techniques, this accuracy matching may be based on the fact that the least-squares functional itself serves as an error measure. Computational experiments conducted for a realistic water table recharge problem illustrate the effectiveness of this approach.