Robust approximation error estimates and multigrid solvers for isogeometric multi-patch discretizations

TL;DR

This paper extends robust approximation error estimates and multigrid solvers from single-patch to multi-patch isogeometric discretizations, enabling efficient PDE solutions on complex geometries composed of multiple patches.

Contribution

It introduces the first extension of robust error estimates and multigrid methods to multi-patch isogeometric analysis, addressing practical complex geometries.

Findings

Robust error estimates are valid for multi-patch geometries.

Multigrid solvers maintain convergence rates in multi-patch settings.

The methods are applicable to real-world CAD models.

Abstract

In recent publications, the author and his coworkers have shown robust approximation error estimates for B-splines of maximum smoothness and have proposed multigrid methods based on them. These methods allow to solve the linear system arizing from the discretization of a partial differential equation in Isogeometric Analysis in a single-patch setting with convergence rates that are provably robust both in the grid size and the spline degree. In real-world problems, the computational domain cannot be nicely represented by just one patch. In computer aided design, such domains are typically represented as a union of multiple patches. In the present paper, we extend the approximation error estimates and the multigrid solver to this multi-patch case.