Isogeometric collocation with smooth mixed degree splines over planar multi-patch domains

TL;DR

This paper introduces a new isogeometric collocation method using smooth mixed degree splines for solving PDEs on multi-patch geometries, reducing degrees of freedom and extending to curved domains.

Contribution

It develops a novel spline space construction for efficient PDE solving on multi-patch domains with curved boundaries, using mixed degree splines and new collocation points.

Findings

Reduced degrees of freedom compared to traditional high-degree spline spaces

Effective solution of Poisson and biharmonic equations on complex geometries

Extension of the method to curved multi-patch domains

Abstract

We present a novel isogeometric collocation method for solving the Poisson's and the biharmonic equation over planar bilinearly parameterized multi-patch geometries. The proposed approach relies on the use of a modified construction of the C^s-smooth mixed degree isogeometric spline space [20] for s=2 and s=4 in case of the Poisson's and the biharmonic equation, respectively. The adapted spline space possesses the minimal possible degree p=s+1 everywhere on the multi-patch domain except in a small neighborhood of the inner edges and of the vertices of patch valency greater than one where a degree p=2s+1 is required. This allows to solve the PDEs with a much lower number of degrees of freedom compared to employing the C^s-smooth spline space [29] with the same high degree p=2s+1 everywhere. To perform isogeometric collocation with the smooth mixed degree spline functions, we introduce and study two different sets of collocation points, namely first a generalization of the standard Greville points to the set of mixed degree Greville points and second the so-called mixed degree superconvergent points. The collocation method is further extended to the class of bilinear-like G^s multi-patch parameterizations [26], which enables the modeling of multi-patch domains with curved boundaries, and is finally tested on the basis of several numerical examples.