Spectral analysis and spectral symbol of matrices in isogeometric collocation methods

A linear full elliptic second order Partial Dierential Equation (PDE), defined on a d-dimensional domain , is approximated by the isogeometric Galerkin method based on uniform tensor-product Bsplines of degrees (p1;:::; pd). The considered approximation process leads to a d-level stiness matrix, banded in a multilevel sense. This matrix is close to a d-level Toeplitz structure when the PDE coecients are constant and the physical domain is just the hypercube (0; 1) d without using any geometry map. In such a simplified case, a detailed spectral analysis of the stiness matrices has been carried out in a previous work. In this paper, we complete the picture by considering non-constant PDE coecients and an arbitrary domain , parameterized with a non-trivial geometry map. We compute and study the spectral symbol of the related stiness matrices. This symbol describes the asymptotic eigenvalue distribution when the fineness parameters tend to zero (so that the matrix-size tends to infinity). The mathematical technique used for computing the symbol is based on the theory of Generalized Locally Toeplitz (GLT) sequences.

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

[2]  T. Hughes,et al.  ISOGEOMETRIC COLLOCATION METHODS , 2010 .

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

[4]  Eugene E. Tyrtyshnikov,et al.  Spectra of multilevel toeplitz matrices: Advanced theory via simple matrix relationships , 1998 .

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

[6]  Carlo Garoni,et al.  A general tool for determining the asymptotic spectral distribution of Hermitian matrix-sequences , 2015 .

[7]  Seymour V. Parter,et al.  Preconditioning Chebyshev Spectral Collocation by Finite-Difference Operators , 1997 .

[8]  Hendrik Speleers,et al.  Robust and optimal multi-iterative techniques for IgA Galerkin linear systems , 2015 .

[9]  W. Rudin.  Real and complex analysis , 1968 .

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

[11]  S. Serra Capizzano,et al.  Distribution results on the algebra generated by Toeplitz sequences: a finite-dimensional approach , 2001 .

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

[13]  Stefano Serra,et al.  On the extreme eigenvalues of hermitian (block) toeplitz matrices , 1998 .

[14]  T. Hughes,et al.  Efficient quadrature for NURBS-based isogeometric analysis , 2010 .

[15]  Carl de Boor,et al.  A Practical Guide to Splines , 1978, Applied Mathematical Sciences.

[16]  Paolo Tilli,et al.  Locally Toeplitz sequences: spectral properties and applications , 1998 .

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

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

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

[20]  S. Serra-Capizzano,et al.  A Note on Antireflective Boundary Conditions and Fast Deblurring Models , 2003, SIAM J. Sci. Comput..

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

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

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

[24]  Weiwei Sun.  Spectral Analysis of Hermite Cubic Spline Collocation Systems , 1999 .

[25]  A. Böttcher,et al.  Introduction to Large Truncated Toeplitz Matrices , 1998 .

[26]  L. Hörmander,et al.  Pseudo-differential Operators and Non-elliptic Boundary Problems , 1966 .

[27]  R. Bhatia.  Matrix Analysis , 1996 .

[28]  Carlo Garoni.  Structured matrices coming from PDE approximation theory: spectral analysis, spectral symbol and design of fast iterative solvers. , 2015 .

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

[30]  Alessandro Reali,et al.  Isogeometric collocation: Cost comparison with Galerkin methods and extension to adaptive hierarchical NURBS discretizations , 2013 .

[31]  Stefano Serra Capizzano,et al.  On the Asymptotic Spectrum of Finite Element Matrix Sequences , 2007, SIAM J. Numer. Anal..

[32]  Carlo Garoni,et al.  Tools for Determining the Asymptotic Spectral Distribution of non-Hermitian Perturbations of Hermitian Matrix-Sequences and Applications , 2015 .

[33]  Stefano Serra Capizzano,et al.  Asymptotic Zero Distribution of Orthogonal Polynomials with Discontinuously Varying Recurrence Coefficients , 2001, J. Approx. Theory.

[34]  A. Quarteroni.  Numerical Models for Differential Problems , 2009 .

[35]  S. Capizzano.  Spectral behavior of matrix sequences and discretized boundary value problems , 2001 .

[36]  Stefano Serra Capizzano,et al.  Spectral Analysis and Spectral Symbol of d-variate $\mathbb Q_{\boldsymbol p}$ Lagrangian FEM Stiffness Matrices , 2015, SIAM J. Matrix Anal. Appl..

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

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