Multigrid solvers for isogeometric discretizations of the second biharmonic problem

TL;DR

This paper introduces a multigrid solver for the second biharmonic problem within Isogeometric Analysis, proving convergence without regularity assumptions and demonstrating efficiency through numerical experiments.

Contribution

The paper extends multigrid convergence analysis to the second biharmonic problem in IgA without regularity assumptions, showing independence from spline degree and zero-order term scaling.

Findings

Convergence rate is independent of spline degree and zero-order term.

Multigrid solver achieves efficient convergence in numerical experiments.

Theoretical bounds depend linearly on the number of levels, logarithmically on grid size.

Abstract

We develop a multigrid solver for the second biharmonic problem in the context of Isogeometric Analysis (IgA), where we also allow a zero-order term. In a previous paper, the authors have developed an analysis for the first biharmonic problem based on Hackbusch's framework. This analysis can only be extended to the second biharmonic problem if one assumes uniform grids. In this paper, we prove a multigrid convergence estimate using Bramble's framework for multigrid analysis without regularity assumptions. We show that the bound for the convergence rate is independent of the scaling of the zero-order term and the spline degree. It only depends linearly on the number of levels, thus logarithmically on the grid size. Numerical experiments are provided which illustrate the convergence theory and the efficiency of the proposed multigrid approaches.