Spectral analysis and spectral symbol of matrices in isogeometric Galerkin methods

Garoni, Carlo; Manni, Carla; Serra-Capizzano, Stefano; Sesana, Debora; Speleers, Hendrik

doi:10.1090/mcom/3143

Spectral analysis and spectral symbol of matrices in isogeometric Galerkin methods
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by Carlo Garoni, Carla Manni, Stefano Serra-Capizzano, Debora Sesana and Hendrik Speleers;

Math. Comp. 86 (2017), 1343-1373

DOI: https://doi.org/10.1090/mcom/3143

Published electronically: August 3, 2016

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Abstract:

A linear full elliptic second-order Partial Differential Equation (PDE), defined on a $d$-dimensional domain $\Omega$, is approximated by the isogeometric Galerkin method based on uniform tensor-product B-splines of degrees $(p_1,\ldots ,p_d)$. The considered approximation process leads to a $d$-level stiffness matrix, banded in a multilevel sense. This matrix is close to a $d$-level Toeplitz structure if the PDE coefficients are constant and the physical domain $\Omega$ is the hypercube $(0,1)^d$ without using any geometry map. In such a simplified case, a detailed spectral analysis of the stiffness matrices has already been carried out in a previous work. In this paper, we complete the picture by considering non-constant PDE coefficients and an arbitrary domain $\Omega$, parameterized with a non-trivial geometry map. We compute and study the spectral symbol of the related stiffness 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 tool used for computing the symbol is the theory of Generalized Locally Toeplitz (GLT) sequences.

References

Robert A. Adams and John J. F. Fournier, Sobolev spaces, 2nd ed., Pure and Applied Mathematics (Amsterdam), vol. 140, Elsevier/Academic Press, Amsterdam, 2003. MR 2424078
Bernhard Beckermann and Stefano Serra-Capizzano, On the asymptotic spectrum of finite element matrix sequences, SIAM J. Numer. Anal. 45 (2007), no. 2, 746–769. MR 2300295, DOI 10.1137/05063533X
Carl de Boor, A practical guide to splines, Revised edition, Applied Mathematical Sciences, vol. 27, Springer-Verlag, New York, 2001. MR 1900298
Albrecht Böttcher and Bernd Silbermann, Introduction to large truncated Toeplitz matrices, Universitext, Springer-Verlag, New York, 1999. MR 1724795, DOI 10.1007/978-1-4612-1426-7
J. A. Cottrell, T. J. R. Hughes, and Y. Bazilevs, Isogeometric Analysis: Toward Integration of CAD and FEA, John Wiley & Sons, 2009.
M. Donatelli, C. Garoni, C. Manni, S. Serra-Capizzano, and H. Speleers, Symbol-based multigrid methods for Galerkin B-spline isogeometric analysis, Tech. Report TW650 (2014), Dept. Computer Science, KU Leuven. Available online at https://lirias.kuleuven.be/handle/ 123456789/461954.
Marco Donatelli, Carlo Garoni, Carla Manni, Stefano Serra-Capizzano, and Hendrik Speleers, Robust and optimal multi-iterative techniques for IgA Galerkin linear systems, Comput. Methods Appl. Mech. Engrg. 284 (2015), 230–264. MR 3310285, DOI 10.1016/j.cma.2014.06.001
Marco Donatelli, Carlo Garoni, Carla Manni, Stefano Serra-Capizzano, and Hendrik Speleers, Robust and optimal multi-iterative techniques for IgA collocation linear systems, Comput. Methods Appl. Mech. Engrg. 284 (2015), 1120–1146. MR 3310320, DOI 10.1016/j.cma.2014.11.036
Marco Donatelli, Carlo Garoni, Carla Manni, Stefano Serra-Capizzano, and Hendrik Speleers, Spectral analysis and spectral symbol of matrices in isogeometric collocation methods, Math. Comp. 85 (2016), no. 300, 1639–1680. MR 3471103, DOI 10.1090/mcom/3027
C. Garoni, Structured matrices coming from PDE approximation theory: spectral analysis, spectral symbol and design of fast iterative solvers, Ph.D. Thesis in Mathematics of Computation, University of Insubria, Como, Italy (2015). Available online at http://hdl.handle.net/ 10277/568.
Carlo Garoni, Carla Manni, Francesca Pelosi, Stefano Serra-Capizzano, and Hendrik Speleers, On the spectrum of stiffness matrices arising from isogeometric analysis, Numer. Math. 127 (2014), no. 4, 751–799. MR 3229992, DOI 10.1007/s00211-013-0600-2
C. Garoni and S. Serra-Capizzano, The theory of Generalized Locally Toeplitz sequences: A review, an extension, and a few representative applications, Tech. Report 2015-023 (2015), Dept. Information Technology, Uppsala University. Available online at http://www.it.uu.se/ research/publications/reports/2015-023/. This is the preliminary version of: C. Garoni and S. Serra-Capizzano, Generalized Locally Toeplitz Sequences: Theory and Applications, Springer Monograph (in preparation).
Carlo Garoni, Stefano Serra-Capizzano, and Debora Sesana, Tools for determining the asymptotic spectral distribution of non-Hermitian perturbations of Hermitian matrix-sequences and applications, Integral Equations Operator Theory 81 (2015), no. 2, 213–225. MR 3299836, DOI 10.1007/s00020-014-2157-6
Carlo Garoni, Stefano Serra-Capizzano, and Debora Sesana, Spectral analysis and spectral symbol of $d$-variate $\Bbb {Q}_p$ Lagrangian FEM stiffness matrices, SIAM J. Matrix Anal. Appl. 36 (2015), no. 3, 1100–1128. MR 3376130, DOI 10.1137/140976480
Carlo Garoni, Stefano Serra-Capizzano, and Paris Vassalos, A general tool for determining the asymptotic spectral distribution of Hermitian matrix-sequences, Oper. Matrices 9 (2015), no. 3, 549–561. MR 3399336, DOI 10.7153/oam-09-33
Leonid Golinskii and Stefano Serra-Capizzano, The asymptotic properties of the spectrum of nonsymmetrically perturbed Jacobi matrix sequences, J. Approx. Theory 144 (2007), no. 1, 84–102. MR 2287378, DOI 10.1016/j.jat.2006.05.002
Lars Hörmander, Pseudo-differential operators and non-elliptic boundary problems, Ann. of Math. (2) 83 (1966), 129–209. MR 233064, DOI 10.2307/1970473
Alfio Quarteroni, Numerical models for differential problems, MS&A. Modeling, Simulation and Applications, vol. 2, Springer-Verlag Italia, Milan, 2009. Translated from the 4th (2008) Italian edition by Silvia Quarteroni. MR 2522375, DOI 10.1007/978-88-470-1071-0
Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR 924157
S. Serra Capizzano, Distribution results on the algebra generated by Toeplitz sequences: a finite-dimensional approach, Linear Algebra Appl. 328 (2001), no. 1-3, 121–130. MR 1823514, DOI 10.1016/S0024-3795(00)00311-6
Stefano Serra Capizzano, Spectral behavior of matrix sequences and discretized boundary value problems, Linear Algebra Appl. 337 (2001), 37–78. MR 1856851, DOI 10.1016/S0024-3795(01)00335-4
S. Serra-Capizzano, A note on antireflective boundary conditions and fast deblurring models, SIAM J. Sci. Comput. 25 (2003/04), no. 4, 1307–1325. MR 2045058, DOI 10.1137/S1064827502410244
S. Serra Capizzano, Generalized locally Toeplitz sequences: spectral analysis and applications to discretized partial differential equations, Linear Algebra Appl. 366 (2003), 371–402. Special issue on structured matrices: analysis, algorithms and applications (Cortona, 2000). MR 1987730, DOI 10.1016/S0024-3795(02)00504-9
Stefano Serra-Capizzano, The GLT class as a generalized Fourier analysis and applications, Linear Algebra Appl. 419 (2006), no. 1, 180–233. MR 2263117, DOI 10.1016/j.laa.2006.04.012
Stefano Serra Capizzano and Cristina Tablino Possio, Analysis of preconditioning strategies for collocation linear systems, Linear Algebra Appl. 369 (2003), 41–75. MR 1988478, DOI 10.1016/S0024-3795(02)00719-X
P. Tilli, Locally Toeplitz sequences: spectral properties and applications, Linear Algebra Appl. 278 (1998), no. 1-3, 91–120. MR 1637331, DOI 10.1016/S0024-3795(97)10079-9
Paolo Tilli, A note on the spectral distribution of Toeplitz matrices, Linear and Multilinear Algebra 45 (1998), no. 2-3, 147–159. MR 1671591, DOI 10.1080/03081089808818584
Evgenij E. Tyrtyshnikov, A unifying approach to some old and new theorems on distribution and clustering, Linear Algebra Appl. 232 (1996), 1–43. MR 1366576, DOI 10.1016/0024-3795(94)00025-5
E. E. Tyrtyshnikov and N. L. Zamarashkin, Spectra of multilevel Toeplitz matrices: advanced theory via simple matrix relationships, Linear Algebra Appl. 270 (1998), 15–27. MR 1484073
H. K. Versteeg and W. Malalasekera, An Introduction to Computational Fluid Dynamics: The Finite Volume Method, Second Edition, Pearson Education Limited, 2007.