An efficient multigrid solver for isogeometric analysis

TL;DR

This paper introduces a multigrid solver for isogeometric analysis that achieves robust convergence independent of spline degree, using overlapping Schwarz smoothers and local Fourier analysis to optimize performance.

Contribution

The paper proposes a novel multigrid approach with adaptive Schwarz smoothers for IGA, demonstrating robustness and efficiency across different spline degrees and discretizations.

Findings

Convergence independent of spline degree and discretization parameters.

Effective multigrid V-cycle with one pre-smoothing step.

Numerical experiments confirm solver efficiency on complex 2D domains.

Abstract

The design of fast solvers for isogeometric analysis is receiving a lot of attention due to the challenge that offers to find an algorithm with a robust convergence with respect to the spline degree. Here, we analyze the application of geometric multigrid methods to this type of discretizations, and we propose a multigrid approach based on overlapping multiplicative Schwarz methods as smoothers. The size of the blocks considered within these relaxation procedures is adapted to the spline degree. A simple multigrid V-cycle with only one step of pre-smoothing results to be a very efficient algorithm, whose convergence is independent on the spline degree and the spatial discretization parameter. Local Fourier analysis is shown to be very useful for the understanding of the problems encountered in the design of a robust multigrid method for IGA, and it is performed to support the good convergence properties of the proposed solver. In fact, an analysis for any spline degree and an arbitrary size of the blocks within the Schwarz smoother is presented for the one-dimensional case. The efficiency of the solver is also demonstrated through several numerical experiments, including a two-dimensional problem on a nontrivial computational domain.