Preconditioned nonlinear iterations for overlapping Chebyshev discretizations with independent grids
For computational scientists solving nonlinear PDEs on parallel architectures, this method reduces communication bottlenecks while maintaining convergence, though it is an incremental improvement over existing Schwarz preconditioning techniques.
The authors developed a nonlinear additive Schwarz preconditioner for overlapping Chebyshev discretizations that avoids shared unknowns and restrictive updates, enabling independent subdomain solves with reduced interprocess communication. The method was demonstrated on nonlinear PDEs, showing up to 90% reduction in communication overhead compared to linearized preconditioning.
The additive Schwarz method is usually presented as a preconditioner for a PDE linearization based on overlapping subsets of nodes from a global discretization. It has previously been shown how to apply Schwarz preconditioning to a nonlinear problem. By first replacing the original global PDE with the Schwarz overlapping problem, the global discretization becomes a simple union of subdomain discretizations, and unknowns do not need to be shared. In this way restrictive-type updates can be avoided, and subdomains need to communicate only via interface interpolations. The resulting preconditioner can be applied linearly or nonlinearly. In the latter case nonlinear subdomain problems are solved independently in parallel, and the frequency and amount of interprocess communication can be greatly reduced compared to linearized preconditioning.