A domain decomposition preconditioner for the nonsymmetric but positive definite linear systems produced by stabilized finite element discretization of three-dimensional advection-diffusion problems, whose coarse space is chosen adaptively but kept small. The subdomain Schur complements are split into symmetric and skew-symmetric parts, the convergence estimate is reduced to bounding the jump operator in the norm of the assembled symmetric part, and that bound is turned into separate face and edge generalized eigenvalue problems whose above-threshold eigenvectors supply the primal constraints. The distinguishing choice is on the edges: instead of building the edge eigenproblem the same way as the face one, the vertex and face constraints already selected are held fixed and the edge Schur complement is formed after eliminating everything except the edge interior and those prior constraints, with the constraint vectors assembled through singular value and QR decompositions. The edge eigenproblems become larger and cost more factorization work, but they parallelize across edges whereas the coarse problem is solved serially, so trading a slightly higher iteration count for a much smaller coarse space is a net win in wall-clock time. A bound independent of subdomain diameter and mesh size, depending only on the threshold, is proved and fed into the standard GMRES residual-decay estimate; experiments with viscosity contrasts of seven orders of magnitude, random viscosity fields, and irregular graph-partitioned subdomains show the largest gains exactly where the geometry and coefficients are worst.
Peng, Shu, Wang and Zhong build a balancing-domain-decomposition-by-constraints (BDDC) preconditioner with an adaptive coarse space for the nonsymmetric but positive definite (NSPD) linear systems th…