Overlapping Schwarz preconditioners for isogeometric discretizations of acoustic wave problems

Abstract

The aim of this work is to construct and analyze two-level overlapping additive Schwarz (OAS) preconditioners for isogeometric discretizations of the acoustic wave equation with absorbing boundary conditions. Both Collocation and Galerkin isogeometric methods are employed for space discretization, while time advancing is performed by means of a Newmark implicit scheme. The linear systems to be solved at each time step are ill conditioned, especially in case of highly regular splines, thus their solution requires the use of effective preconditioners. Two-level OAS solvers consist of partitioning the domain into overlapping subdomains, solving independent local problems on each subdomain and an additional coarse problem associated with the subdomain mesh. Several two-dimensional numerical results validate our theoretical estimates, showing the scalability and quasi-optimality of the algorithms proposed. We also investigate numerically the robustness of the OAS preconditioners with respect to the spline polynomial degree, the spline regularity and the overlap parameter.