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

2017/11/29 by Fabio Roman, Roman, Fabio
Computer Science · Engineering · Mathematics · #Advanced Numerical Analysis Techniques #FOS: Mathematics #Iterative Methods for Nonlinear Equations #Numerical Analysis (math.NA) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.1711.10917

openalex publication_date 2017/11/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• The GLT class as a generalized Fourier analysis and applications
• Convergence Analysis of Krylov Subspace Iterations with Methods from Potential Theory
• Generalized locally Toeplitz sequences: spectral analysis and applications to discretized partial differential equations
• Superlinear Convergence of Conjugate Gradients

Related

• Complementary results for the spectral analysis of matrices in Galerkin methods with GB-splines
• Collocation and Mass Matrix in Least-squares Isogeometric Analysis
• Multi-resolution Isogeometric Analysis -- Efficient adaptivity utilizing the multi-patch structure
• Adaptive isogeometric methods with C¹ (truncated) hierarchical splines on planar multi-patch domains
• Isogeometric collocation with smooth mixed degree splines over planar multi-patch domains