Multigrid Algorithms for the Solution of Linear Complementarity Problems Arising from Free Boundary Problems

Abstract

Several free boundary problems (including saturated-unsaturated flow through porous dams, elastic-plastic torsion and cavitating journal bearings) can be formulated as linear complementarity problems of the following type: Find a nonnegative function u which satisfies prescribed boundary conditions on a given domain and which, furthermore, satisfies a linear elliptic equation at each point of the domain where u is greater than zero. We show that the multigrid FAS algorithm, which was developed by Brandt to solve boundary value problems for elliptic partial differential equations, can easily be adapted to handle linear complementarity problems. For large problems, the resulting algorithm, PFAS (projected full approximation scheme) is significantly faster than previous single-grid algorithms, since the computation time is proportional to the number of grid points on the finest grid. We then introduce two further multigrid algorithms, PFASMD and PFMG. PFASMD is a modification of PFAS which is considerably faster than PFAS. Using PFMG (projected full multigrid) it is possible to solve a linear complementarity problem to within truncation error using less work than the equivalent of seven Gauss–Seidel sweeps on the finest grid.