We present the full approximation scheme constraint decomposition (FASCD)
multilevel method for solving variational inequalities (VIs). FASCD is a common
extension of both the full approximation scheme (FAS) multigrid technique for
nonlinear partial differential equations, due to A.~Brandt, and the constraint
decomposition (CD) method introduced by X.-C.~Tai for VIs arising in
optimization. We extend the CD idea by exploiting the telescoping nature of
certain function space subset decompositions arising from multilevel mesh
hierarchies. When a reduced-space (active set) Newton method is applied as a
smoother, with work proportional to the number of unknowns on a given mesh
level, FASCD V-cycles exhibit nearly mesh-independent convergence rates, and
full multigrid cycles are optimal solvers. The example problems include
differential operators which are symmetric linear, nonsymmetric linear, and
nonlinear, in unilateral and bilateral VI problems.Comment: 25 pages, 9 figure