A Full Approximation Scheme Multilevel Method for Nonlinear Variational Inequalities

SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page A2421-A2444, August 2024.
Abstract. We present the full approximation scheme constraint decomposition (FASCD) multilevel method for solving variational inequalities (VIs). FASCD is a joint extension of both the full approximation scheme 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 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. The full multigrid cycle version is an optimal solver. The example problems include differential operators which are symmetric linear, nonsymmetric linear, and nonlinear, in unilateral and bilateral VI problems. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://bitbucket.org/pefarrell/fascd/, where the software used to produce the results in section 8 is archived at tag v1.0, and at https://doi.org/10.5281/zenodo.10476845 or in the supplementary materials (pefarrell-fascd-6407e9f547d6.zip [225KB]). The authors used Firedrake master revision c5e939dde.