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

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.

Explicit error estimates for spline approximation of arbitrary smoothness in isogeometric analysis

“…The p-dependence has also been strengthened for the arbitrarily smooth case in Corollary 5. Motivated by their use in the analysis of fast iterative solvers for linear systems arising from spline discretization methods [15], we also provide error estimates for approximation in suitable reduced spline spaces; see Theorems 5 and 6.…”

“…Let (a, b) = (0, 1), and assume a uniform knot sequence Ξ and h 1. Then, keeping the dimension formula (15) in mind, the first inequality in Remark 3 can be rephrased as: for any q = , . .…”

“…Besides their prominent interest for analyzing convergence under different kinds of refinements, error estimates for approximation in suitable reduced spline spaces play a less evident but still pivotal role in other aspects of IGA discretizations, such as the design of fast iterative (multigrid) solvers for the resulting linear systems [15,24]. The convergence rate of fast iterative solvers should ideally be independent of all the parameters involved, and so their explicit impact on the estimates is important to understand.…”

BPX preconditioners for isogeometric analysis using analysis-suitable T-splines

“…where the subspaces V k are defined in (13), and thus obtain BPX preconditioners on locally quasiuniform AS T-meshes. [24] proposed a new smoother based on the mass matrix and a boundary correction, that was later improved in [23]. The results in those papers show that on a uniform mesh, the multigrid with the new smoother is more robust, in the sense that convergence is independent of both the mesh size and the spline degree.…”

Inverse mass matrix for isogeometric explicit transient analysis via the method of localized Lagrange multipliers

“…However, this procedure is not completely general and nonseparable mass matrices require a carefully designed preconditioner that adds significant complexity and computational cost to the linear solver. New preconditioning and multigrid techniques with linear cost [17][18][19] could also be considered for that task.…”