A Discovery Algorithm for the Algebraic Construction of Optimized Schwarz Preconditioners

Summary

Optimized Schwarz methods have been developed at the continuous level; in order to obtain optimized transmission conditions, the underlying partial differential equation (PDE) needs to be known. Classical Schwarz methods on the other hand can be used in purely algebraic form, which have made them popular. Their performance can however be inferior compared to that of optimized Schwarz methods. We present in this paper a discovery algorithm, which, based purely on algebraic information, allows us to obtain an optimized Schwarz preconditioner for a large class of numerically discretized elliptic PDEs. The algorithm detects the nature of the elliptic PDE, and then modifies a classical algebraic Schwarz preconditioner at the algebraic level, using existing optimization results from the literature on optimized Schwarz methods. Numerical experiments using elliptic problems discretized by Q ₁-FEM, P ₁-FEM, and FDM demonstrate the algebraic nature and the effectiveness of the discovery algorithm.