Overlapping Schwarz Methods for Isogeometric Analysis Based on Generalized B-splines

We construct and analyze an overlapping Schwarz preconditioner forelliptic problems discretized with isogeometricanalysis. The preconditioner is based on partitioning the domainof the problem into overlapping subdomains, solving localisogeometric problems on these subdomains, and solving anadditional coarse isogeometric problem associated with thesubdomain mesh. We develop an $h$-analysis of the preconditioner,showing in particular that the resulting algorithm is scalable andits convergence rate depends linearly on the ratio betweensubdomain and „overlap sizes” for fixed polynomial degree $p$and regularity $k$ of the basis functions. Numerical results in two-and three-dimensional tests show the good convergence properties of thepreconditioner with respect to the isogeometric discretizationparameters $h, p, k$, number of subdomains $N$, overlap size, andalso jumps in the coefficients of the ellipticoperator.