Solving the triharmonic equation over multi-patch domains using isogeometric analysis

TL;DR

This paper introduces a novel isogeometric analysis framework for solving the triharmonic equation on multi-patch domains, utilizing a globally smooth spline space with local functions that are easy to construct and well-conditioned.

Contribution

It develops a simple, uniform method to construct a globally $C^2$-smooth spline space for multi-patch domains, enabling efficient triharmonic equation solutions.

Findings

The method produces well-conditioned, local basis functions.

Numerical examples demonstrate the approach's effectiveness.

The construction is simple and applicable to various configurations.

Abstract

We present a framework for solving the triharmonic equation over bilinearly parameterized planar multi-patch domains by means of isogeometric analysis. Our approach is based on the construction of a globally $C^2$-smooth isogeometric spline space which is used as discretization space. The generated $C^2$-smooth space consists of three different types of isogeometric functions called patch, edge and vertex functions. All functions are entirely local with a small support, and numerical examples indicate that they are well-conditioned. The construction of the functions is simple and works uniformly for all multi-patch configurations. While the patch and edge functions are given by a closed form representation, the vertex functions are obtained by computing the null space of a small system of linear equations. Several examples demonstrate the potential of our approach for solving the triharmonic equation.