Robust multigrid solvers for the biharmonic problem in isogeometric analysis

TL;DR

This paper introduces two robust multigrid solvers for the biharmonic problem within isogeometric analysis, demonstrating their effectiveness and robustness through theoretical proofs and numerical experiments.

Contribution

It develops and analyzes two multigrid methods tailored for $H^2$-conforming discretizations in isogeometric analysis, including a hybrid approach for enhanced performance.

Findings

Both multigrid methods are robust with respect to grid size.

The mass smoothing method is also robust in spline degree.

Numerical experiments confirm the efficiency and convergence of the proposed methods.

Abstract

In this paper, we develop multigrid solvers for the biharmonic problem in the framework of isogeometric analysis (IgA). In this framework, one typically sets up B-splines on the unit square or cube and transforms them to the domain of interest by a global smooth geometry function. With this approach, it is feasible to set up $H^2$-conforming discretizations. We propose two multigrid methods for such a discretization, one based on Gauss Seidel smoothing and one based on mass smoothing. We prove that both are robust in the grid size, the latter is also robust in the spline degree. Numerical experiments illustrate the convergence theory and indicate the efficiency of the proposed multigrid approaches, particularly of a hybrid approach combining both smoothers.