Robust and optimal multi-iterative techniques for IgA collocation linear systems

Abstract

We consider fast solvers for the large linear systems coming from the Isogeometric Analysis (IgA) collocation approximation based on B-splines of full elliptic d-dimensional Partial Differential Equations (PDEs). We are interested in designing iterative algorithms which are optimal and robust. The former property implies that the computational cost is linear with respect to the number of degrees of freedom (i.e. the matrix size). The latter property means that the convergence rate is completely independent of (or only mildly dependent on) all the relevant parameters: in our setting, we can mention the coefficients of the PDE, the dimensionality d, the geometric map G describing the physical domain, the matrix size (related to the fineness parameters), and the spline degrees (associated with the IgA approximation order). Our approach is based on the spectral symbol, which describes the global eigenvalue behavior of the IgA collocation matrices. It is precisely the spectral information contained in the symbol that is exploited for the design of an optimal and robust multi-iterative solver of multigrid type. Several numerical experiments are presented and discussed in view of recent theoretical findings concerning the symbol and its properties.