Two-grid optimality for Galerkin linear systems based on B-splines

Abstract

A multigrid method for linear systems stemming from the Galerkin B-spline discretization of classical second-order elliptic problems is considered. The spectral features of the involved stiffness matrices, as the fineness parameter h tends to zero, have been deeply studied in previous works, with particular attention to the dependencies of the spectrum on the degree p of the B-splines used in the discretization process. Here, by exploiting this information in connection with $$\tau $$?-matrices, we describe a multigrid strategy and we prove that the corresponding two-grid iterations have a convergence rate independent of h for $$p=1,2,3$$p=1,2,3. For larger p, the proof may be obtained through algebraic manipulations. Unfortunately, as confirmed by the numerical experiments, the dependence on p is bad and hence other techniques have to be considered for large p.