Robust Multigrid for Isogeometric Analysis Based on Stable Splittings of Spline Spaces

TL;DR

This paper introduces a robust multigrid method for isogeometric analysis that effectively handles high-degree splines and boundary effects, ensuring stable and efficient solutions across various dimensions.

Contribution

It proposes a novel stable spline space splitting and a multigrid smoother that together achieve degree- and mesh-independent convergence in isogeometric discretizations.

Findings

Iteration numbers are robust to spline degree and mesh size.

Method scales well with problem dimension.

Numerical examples confirm theoretical robustness.

Abstract

We present a robust and efficient multigrid method for single-patch isogeometric discretizations using tensor product B-splines of maximum smoothness. Our method is based on a stable splitting of the spline space into a large subspace of interior splines which satisfy a robust inverse inequality, as well as one or several smaller subspaces which capture the boundary effects responsible for the spectral outliers which occur in Isogeometric Analysis. We then construct a multigrid smoother based on an additive subspace correction approach, applying a different smoother to each of the subspaces. For the interior splines, we use a mass smoother, whereas the remaining components are treated with suitably chosen Kronecker product smoothers or direct solvers. We prove that the resulting multigrid method exhibits iteration numbers which are robust with respect to the spline degree and the mesh size. Furthermore, it can be efficiently realized for discretizations of problems in arbitrarily high geometric dimension. Some numerical examples illustrate the theoretical results and show that the iteration numbers also scale relatively mildly with the problem dimension.