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

We consider fast solvers for large linear systems arising from the Galerkin approximation based on B-splines of classical d-dimensional elliptic problems, d≥. 1, in the context of isogeometric analysis. Our ultimate goal is to design iterative algorithms with the following two properties. First, their computational cost is optimal, that is linear with respect to the number of degrees of freedom, i.e.the resulting matrix size. Second, they are totally robust, i.e., their convergence speed is substantially independent of all the relevant parameters: in our case, these are the matrix size (related to the fineness parameter), the spline degree (associated to the approximation order), and the dimensionality d of the problem. We review several methods like PCG, multigrid, multi-iterative algorithms, and we show how their numerical behavior (in terms of convergence speed) can be understood through the notion of spectral distribution, i.e.through a compact symbol which describes the global eigenvalue behavior of the considered stiffness matrices. As a final step, we show how we can design an optimal and totally robust multi-iterative method, by taking into account the analytic features of the symbol. A wide variety of numerical experiments, few open problems and perspectives are presented and critically discussed.