A Robust Multigrid Method for Isogeometric Analysis using Boundary Correction

TL;DR

This paper develops a multigrid solver for isogeometric analysis that maintains robust convergence rates even as the spline degree increases, leveraging recent theoretical advances in B-spline approximation.

Contribution

It introduces a boundary correction technique that ensures multigrid methods are robust for high-degree B-spline discretizations in isogeometric analysis.

Findings

Achieves convergence rates independent of spline degree

Demonstrates effectiveness through numerical experiments

Provides theoretical guarantees for robustness

Abstract

We consider geometric multigrid methods for the solution of linear systems arising from isogeometric discretizations of elliptic partial differential equations. For classical finite elements, such methods are well known to be fast solvers showing optimal convergence behavior. However, the naive application of multigrid to the isogeometric case results in significant deterioration of the convergence rates if the spline degree is increased. Recently, a robust approximation error estimate and a corresponding inverse inequality for B-splines of maximum smoothness have been shown, both with constants independent of the spline degree. We use these results to construct multigrid solvers for discretizations based on B-splines with maximum smoothness which exhibit robust convergence rates.