Spectral analysis of matrices in collocation methods based on generalized B-splines with high smoothness

TL;DR

This paper analyzes the spectral properties of matrices from isogeometric collocation methods using generalized B-splines to efficiently solve elliptic PDEs with variable coefficients and complex geometries.

Contribution

It provides a spectral analysis of matrices in collocation methods based on GB-splines, advancing understanding of their numerical properties in isogeometric analysis.

Findings

Spectral properties depend on spline smoothness and geometry.

Results improve understanding of matrix behavior in collocation methods.

Analysis aids in designing more efficient isogeometric solvers.

Abstract

Although Galerkin discretizations have been intensively employed in the IgA context, an efficient implementation requires special numerical quadrature rules when constructing the system of equations. To avoid this issue, isogeometric collocation methods have been recently introduced, giving origin to a topic of study which is proceeding almost parallel to the Galerkin evolution. In this paper we analyze the spectral properties of the matrices arising from isogeometric collocation methods based on GB-splines to approximate an elliptic PDE with variable coefficients and a generic geometry map.