Hybrid TBETI domain decomposition for huge 2D scalar variational inequalities

The unpreconditioned H-TFETI-DP (hybrid total finite element tearing and interconnecting dual-primal) domain decomposition method introduced by Klawonn and Rheinbach turned out to be an effective solver for variational inequalities discretized by huge structured grids. The basic idea is to decompose the domain into non-overlapping subdomains, interconnect some adjacent subdomains into clusters on a primal level, and enforce the continuity of the solution across both the subdomain and cluster interfaces by Lagrange multipliers. After eliminating the primal variables, we get a reasonably conditioned quadratic programming (QP) problem with bound and equality constraints. Here, we first reduce the continuous problem to the subdomains' boundaries, then discretize it using the boundary element method, and finally interconnect the subdomains by the averages of adjacent edges. The resulting QP problem in multipliers with a small coarse grid is solved by specialized QP algorithms with optimal complexity. The method can be considered as a three-level multigrid with the coarse grids split between primal and dual variables. Numerical experiments illustrate the efficiency of the presented H-TBETI-DP (hybrid total boundary element tearing and interconnecting dual-primal) method and nice spectral properties of the discretized Steklov–Poincaré operators as compared with their finite element counterparts.