On Robust Multigrid Methods for Non-Smooth Variational Problems

Abstract

We consider the fast solution of large piecewise smooth minimization problems as resulting from the approximation of elliptic free boundary problems. The most delicate question in constructing a multigrid method for a nonlinear non-smooth problem is how to represent the nonlinearity on the coarse grids. This process usually involves some kind of linearization. The basic idea of monotone multigrid methods to be presented here is first to select a neighborhood of the actual smoothed iterate in which a linearization is possible and then to constrain the coarse grid correction to this neighborhood. Such a local linearization allows to control the local corrections at each coarse grid node in such a way that the energy functional is monotonically decreasing. This approach leads to globally convergent schemes which are robust with respect to local singularities of the given problem. The numerical performance is illustrated by approximating the well-known Barenblatt solution of the porous medium equation.

The author gratefully acknowledges the hospitality of P. Deuflhard and his staff at the Konrad-Zuse-Center Berlin during the preparation of this manuscript. The work was supported by a Konrad-Zuse-Fellowship.