Algebraic Multilevel Preconditioning in Isogeometric Analysis: Construction and Numerical Studies

Abstract: We present algebraic multilevel iteration (AMLI) methods for isogeometric discretization of scalar second order elliptic problems. The construction of coarse grid operators and hierarchical complementary operators are given. Moreover, for a uniform mesh on a unit interval, the explicit representation of B-spline basis functions for a fixed mesh size $h$ is given for $p=2,3,4$ and for $C^{0}$- and $C^{p-1}$-continuity. The presented methods show $h$- and (almost) $p$-independent convergence rates. Supporting numerical results for convergence factor and iterations count for AMLI cycles ($V$-, linear $W$-, nonlinear $W$-) are provided. Numerical tests are performed, in two-dimensions on square domain and quarter annulus, and in three-dimensions on quarter thick ring.