Algebraic Multilevel Preconditioning in Isogeometric Analysis: Construction and Numerical Studies

TL;DR

This paper develops algebraic multilevel iteration methods for isogeometric analysis, providing explicit basis function representations and demonstrating convergence rates that are largely independent of mesh size and polynomial degree through numerical experiments.

Contribution

It introduces AMLI methods tailored for isogeometric discretizations, including explicit basis function formulas and comprehensive numerical validation.

Findings

Convergence rates are nearly independent of mesh size and polynomial degree.

Explicit B-spline basis functions are derived for specific degrees and continuities.

Numerical results confirm the effectiveness of AMLI cycles in various dimensions.

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.