Practical unstructured splines: Algorithms, multi-patch spline spaces, and some applications to numerical analysis

• Published in: Journal of computational physics 2022-12, Vol.471, p.111625, Article 111625
• Main Authors: Frambati, Stefano, Barucq, Hélène, Calandra, Henri, Diaz, Julien
• Format: Article
• Language: English
• Subjects: Discontinuous Galerkin; Isogeometric analysis; Mathematics; Multi-patch; Numerical Analysis; Simplex splines
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2022.111625

•Efficient algorithms for unstructured simplex spline spaces.•Multi-patch spline domains including non-convex, non-simply-connected patches.•Fully unstructured multi-patch discontinuous Galerkin-isogeometric analysis.•Usual discontinuous Galerkin basis is obtained as a special case.•Block-diagonal mass matrix, useful for time-explicit hyperbolic problems. In this work, we show how some recent advances on simplex spline spaces can be used to construct a polynomial-reproducing space of unstructured splines on multi-patch domains of arbitrary shape and topology. The traces of these functions on the subdomain boundaries reproduce the usual traces of standard polynomial bases used in discontinuous Galerkin (DG) approximations, allowing to borrow many theoretical and practical tools from these methods. Concurrently, we recast some theoretical results on the construction and evaluation of spaces of simplex splines into an explicit, algorithmic form. Together, these efforts allow to formulate a practical, efficient and fully unstructured multi-patch discontinuous Galerkin - isogeometric analysis (DG-IGA) scheme that bridges the gap between some current multi-patch isogeometric analysis (IGA) approaches and the more traditional mesh-based interior penalty discontinuous Galerkin (IPDG) method. We briefly discuss the advantages of this unified framework for time-explicit hyperbolic problems, and we present some interesting numerical examples using the acoustic wave equation.