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

We present a first step towards the spectral analysis of matrices arising from IgA Galerkin methods based on hyperbolic and trigonometric GB-splines. Second order differential problems with constant coefficients are considered and discretized by means of sequences of both nested and non-nested spline spaces. We prove that there always exists an asymptotic eigenvalue distribution which can be compactly described by a symbol, just like in the polynomial case. There is a complete similarity between the symbol expressions in the hyperbolic, trigonometric and polynomial cases. This results in similar spectral features of the corresponding matrices. We also analyze the IgA discretization based on trigonometric GB-splines for the eigenvalue problem related to the univariate Laplace operator. We prove that, for non-nested spaces, the phase parameter can be exploited to improve the spectral approximation with respect to the polynomial case. As part of the analysis, we derive several Fourier properties of cardinal GB-splines.

[1]  Arno B. J. Kuijlaars,et al.  Superlinear Convergence of Conjugate Gradients , 2001, SIAM J. Numer. Anal..

[2]  P. Sattayatham,et al.  GB-splines of arbitrary order , 1999 .

[3]  Guozhao Wang,et al.  Unified and extended form of three types of splines , 2008 .

[4]  Tom Lyche,et al.  On a class of weak Tchebycheff systems , 2005, Numerische Mathematik.

[5]  Alessandro Reali,et al.  Isogeometric Analysis of Structural Vibrations , 2006 .

[6]  Stefano Serra-Capizzano,et al.  The GLT class as a generalized Fourier analysis and applications , 2006 .

[7]  J. M. Peña,et al.  Critical Length for Design Purposes and Extended Chebyshev Spaces , 2003 .

[8]  Hendrik Speleers,et al.  Spectral analysis and spectral symbol of matrices in isogeometric collocation methods , 2015, Math. Comput..

[9]  Charles K. Chui,et al.  An Introduction to Wavelets , 1992 .

[10]  S. Serra Capizzano,et al.  Generalized locally Toeplitz sequences: spectral analysis and applications to discretized partial differential equations , 2003 .

[11]  T. Hughes,et al.  Isogeometric analysis : CAD, finite elements, NURBS, exact geometry and mesh refinement , 2005 .

[12]  Paolo Tilli,et al.  On unitarily invariant norms of matrix-valued linear positive operators , 2002 .

[13]  Hendrik Speleers,et al.  Spectral analysis and spectral symbol of matrices in isogeometric Galerkin methods , 2017, Math. Comput..

[14]  U. Grenander,et al.  Toeplitz Forms And Their Applications , 1958 .

[15]  Carla Manni,et al.  Isogeometric analysis in advection-diffusion problems: Tension splines approximation , 2011, J. Comput. Appl. Math..

[16]  Arno B. J. Kuijlaars,et al.  Convergence Analysis of Krylov Subspace Iterations with Methods from Potential Theory , 2006, SIAM Rev..

[17]  Thomas J. R. Hughes,et al.  Isogeometric Analysis: Toward Integration of CAD and FEA , 2009 .

[18]  Carla Manni,et al.  Generalized B-splines as a tool in Isogeometric Analysis , 2011 .

[19]  Hendrik Speleers,et al.  Symbol-Based Multigrid Methods for Galerkin B-Spline Isogeometric Analysis , 2017, SIAM J. Numer. Anal..

[20]  Giancarlo Sangalli,et al.  Some estimates for h–p–k-refinement in Isogeometric Analysis , 2011, Numerische Mathematik.

[21]  Hendrik Speleers,et al.  Robust and optimal multi-iterative techniques for IgA Galerkin linear systems This is a preprint of a paper published in Comput. Methods Appl. Mech. Engrg. 284 (2015) 230264. , 2015 .

[22]  Leonid Golinskii,et al.  The asymptotic properties of the spectrum of nonsymmetrically perturbed Jacobi matrix sequences , 2007, J. Approx. Theory.

[23]  Hendrik Speleers,et al.  Isogeometric collocation methods with generalized B-splines , 2015, Comput. Math. Appl..

[24]  Stefano Serra Capizzano,et al.  Analysis of preconditioning strategies for collocation linear systems , 2003 .

[25]  Stefano Serra Capizzano,et al.  V-cycle Optimal Convergence for Certain (Multilevel) Structured Linear Systems , 2004, SIAM J. Matrix Anal. Appl..

[26]  Paolo Tilli,et al.  A note on the spectral distribution of toeplitz matrices , 1998 .

[27]  Alessandro Reali,et al.  Finite element and NURBS approximations of eigenvalue, boundary-value, and initial-value problems , 2014 .

[28]  Thierry Blu,et al.  Cardinal exponential splines: part I - theory and filtering algorithms , 2005, IEEE Transactions on Signal Processing.

[29]  Stefano Serra,et al.  The rate of convergence of Toeplitz based PCG methods for second order nonlinear boundary value problems , 1999 .

[30]  Stefano Serra Capizzano,et al.  The rate of convergence of Toeplitz based PCG methods for second order nonlinear boundary value problems , 1999, Numerische Mathematik.

[31]  S. Serra,et al.  Multi-iterative methods , 1993 .

[32]  Hendrik Speleers,et al.  On the spectrum of stiffness matrices arising from isogeometric analysis , 2014, Numerische Mathematik.

[33]  E. E. Tyrtyshnikov.  A unifying approach to some old and new theorems on distribution and clustering , 1996 .

[34]  S. Serra-Capizzano,et al.  The theory of generalized locally Toeplitz sequences : a review, an extension, and a few representative applications , 2017 .