# Isogeometric collocation on planar multi-patch domains

## TL;DR

This paper introduces a simple, globally smooth isogeometric collocation method for solving Poisson's equation on multi-patch domains, analyzing different collocation point choices for stability and convergence.

## Contribution

It develops a $C^2$-smooth discretization space for multi-patch domains and compares two collocation point strategies, including a generalized superconvergent points approach.

## Key findings

- Greville abscissae show suboptimal convergence for odd spline degrees.
- Superconvergent points improve convergence in multi-patch cases.
- The method is uniformly applicable to all multi-patch configurations.

## Abstract

We present an isogeometric framework based on collocation to construct a $C^2$-smooth approximation of the solution of the Poisson's equation over planar bilinearly parameterized multi-patch domains. The construction of the used globally $C^2$-smooth discretization space for the partial differential equation is simple and works uniformly for all possible multi-patch configurations. The basis of the $C^2$-smooth space can be described as the span of three different types of locally supported functions corresponding to the single patches, edges and vertices of the multi-patch domain. For the selection of the collocation points, which is important for the stability and convergence of the collocation problem, two different choices are numerically investigated. The first approach employs the tensor-product Greville abscissae as collocation points, and shows for the multi-patch case the same convergence behavior as for the one-patch case [2], which is suboptimal in particular for odd spline degree. The second approach generalizes the concept of superconvergent points from the one-patch case (cf. [1, 15, 32]) to the multi-patch case. Again, these points possess better convergence properties than Greville abscissae in case of odd spline degree.