Overlapping Schwarz methods for isogeometric analysis based on generalized B-splines

Abstract

Generalized B-splines (GB-splines) are a special class of Tchebycheff B-splines that are smooth piecewise function with sections in more general spaces. GB-splines allow for an exact representation of conic sections as well as transcendental curves and thus they become very attractive for geometrical modeling and numerical simulation. In this paper, we present overlapping Schwarz preconditioners for elliptic problems discretized with isogeometric analysis based on GB-splines. An h-analysis of the proposed preconditioners shows an optimal convergence rate bound that is scalable in the number of subdomains and that is linear in the ratio between subdomain and overlap sizes. Numerical results in twoand three-dimensional tests confirm this analysis and also illustrate the good convergence properties of the preconditioner with respect to the discretization parameters, the domain deformation and the jumping coefficients. (C) 2020 Elsevier B.V. All rights reserved.