, 55:47 | Cite as

Spectral analysis of finite-dimensional approximations of 1d waves in non-uniform grids

  • Davide BianchiEmail author
  • Stefano Serra-Capizzano


We study the gap of discrete spectra of the Laplace operator in 1d for non-uniform meshes by analyzing the corresponding spectral symbol, which allows to show how to design the discretization grid for improving the gap behavior. The main tool is the study of a univariate monotonic version of the spectral symbol, obtained by employing a proper rearrangement via the GLT theory. We treat in detail the case of basic finite-difference approximations. In a second step, we pass to precise approximation schemes, coming from the celebrated Galerkin isogeometric analysis based on B-splines of degree p and global regularity \(C^{p-1}\), and finally we address the case of finite-elements with global regularity \(C^0\) and local polynomial degree p. The surprising result is that the GLT approach allows a unified spectral treatment of the various schemes also in terms of the preservation of the average gap property, which is necessary for the uniform gap property. The analytical results are illustrated by a number of numerical experiments. We conclude by discussing some open problems.


Wave equation Boundary control and observation Finite-differences Finite-elements Isogeometric analysis Velocity of propagation Non-uniform grids versus approximately weakly regular grids Spectral symbol Spectral gap 

Mathematcis Subject Classification

65F10 65N22 15A18 15A12 47B65 



We are grateful to Professor Enrique Zuazua for the time he dedicated to us and to this work, for discussions and illuminating advices. Without his help this paper would have never came to light. A special thank goes to Carlo Garoni for his careful reading and his numerous appropriate suggestions. Finally, we thank a lot the Editor and the Reviewers for their comments and criticisms which helped us both in improving the quality of the presentation and in making the content more complete.


  1. 1.
    Al-Fhaid, A.S., Sesana, D., Serra-Capizzano, S., Ullah, M.Z.: Singular-value (and eigenvalue) distribution and Krylov preconditioning of sequences of sampling matrices approximating integral operators. Numer. Linear Algebra Appl. 21, 722–743 (2014)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Beckermann, B., Serra-Capizzano, S.: On the asymptotic spectrum of finite elements matrices. SIAM J. Numer. Anal. 45, 746–769 (2007)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Di Benedetto, F., Fiorentino, G., Serra-Capizzano, S.: CG preconditioning for Toeplitz matrices. Comput. Math. Appl. 25, 33–45 (1993)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Donatelli, M., Garoni, C., Manni, C., Serra-Capizzano, S., Speleers, H.: Spectral analysis and spectral symbol of matrices in isogeometric collocation methods. Math. Comput. 85, 1639–1680 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Donatelli, M., Mazza, M., Serra-Capizzano, S.: Spectral analysis and structure preserving preconditioners for fractional diffusion equations. J. Comput. Phys. 307, 262–279 (2016)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Ervedoza, S., Marica, A., Zuazua, E.: Numerical meshes ensuring uniform observability of one-dimensional waves: construction and analysis. IMA J. Numer. Anal. 36, 503–542 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Ervedoza, S., Zuazua, E.: The wave equation: control and numerics. In: Control of Partial Differential Equations, Lecture Notes in Mathematics, vol. 2048, pp. 245–339 (2012)Google Scholar
  8. 8.
    Fernández-Cara, E., Zuazua, E.: On the null controllability of the one-dimensional heat equation with BV coefficients. Comput. Appl. Math. 21, 167–190 (2002)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Garoni, C., Speleers, H., Ekström, S.-E., Hughes, T.J.R., Reali, A., Serra-Capizzano, S.: Symbol-based analysis of finite element and isogeometric B-spline discretizations of eigenvalue problems: exposition and review. Arch. Comput. Methods Eng. (2018).
  10. 10.
    Garoni, C., Manni, C., Serra-Capizzano, S., Sesana, D., Speleers, H.: Spectral analysis and spectral symbol of matrices in isogeometric Galerkin methods. Math. Comput. 86, 1343–1373 (2017)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Garoni, C., Mazza, M., Serra-Capizzano, S.: Block generalized locally Toeplitz sequences: from the theory to the applications. Axioms 7(3), 49 (2018)CrossRefGoogle Scholar
  12. 12.
    Garoni, C., Serra-Capizzano, S.: Generalized Locally Toeplitz Sequences: Theory and Applications (Volume I). Springer, Cham (2017)CrossRefGoogle Scholar
  13. 13.
    Garoni, C., Serra-Capizzano. S.: Generalized Locally Toeplitz Sequences: Theory and Applications (Volume II). Springer, Cham, to appearGoogle Scholar
  14. 14.
    Garoni, C., Serra-Capizzano, S., Sesana, D.: Spectral analysis and spectral symbol of \(d\)-variate \(\mathbb{Q}_p\) Lagrangian FEM stiffness matrices. SIAM J. Matrix Anal. Appl. 36, 1100–1128 (2015)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Garoni, C., Serra-Capizzano, S., Sesana, D.: Block generalized locally Toeplitz sequences: topological construction, spectral distribution results, and star-algebra structure. Springer INdAM Series, To appearGoogle Scholar
  16. 16.
    Glowinski, R., Li, C.H.: On the numerical implementation of the Hilbert uniqueness method for the exact boundary controllability of the wave equation. C.R. Acad. Sci. Paris Sér. I Math. 311, 135–142 (1990)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Glowinski, R., Li, C.H., Lions, J.-L.: A numerical approach to the exact boundary controllability of the wave equation (I) Dirichlet controls: description of the numerical methods. Jpn. J. Appl. Math. 7, 1–76 (1990)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Hughes, T.J.R., Evans, J.A., Reali, A.: Finite element and NURBS approximations of eigenvalue, boundary-value, and initial-value problems. Comput. Methods Appl. Mech. Eng. 272, 290–320 (2014)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Infante, J.A., Zuazua, E.: Boundary observability for the space semi discretizations of the 1-d wave equation. Math. Model. Num. Ann. 33, 407–438 (1999)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Ingham, A.E.: Some trigonometrical inequalities with applications to the theory of series. Math. Z. 41, 367–379 (1936)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Lions, J.-L.: Contrôlabilité exacte, Stabilisation et Perturbations de Systèmes Distribués. Tome 1. Contrôlabilité exacte, volume RMA 8. Masson (1988)Google Scholar
  22. 22.
    Macia, F.: Wigner measures in the discrete setting: high-frequency analysis of sampling and reconstruction operators. SIAM J. Math. Anal. 36–2, 347–383 (2004)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Marica, A., Zuazua, E.: On the quadratic finite element approximation of 1-d waves: propagation, observation and control. SIAM J. Numer. Anal. 50, 2744–2777 (2012)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Marica, A., Zuazua, E.: Propagation of 1-D waves in regular discrete heterogeneous media: a Wigner measure approach. Found. Comput. Math. 15, 1571–1636 (2015)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Serra-Capizzano, S.: Generalized locally Toeplitz sequences: spectral analysis and applications to discretized partial differential equations. Linear Algebra Appl. 366, 371–402 (2003)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Serra-Capizzano, S.: The GLT class as a generalized Fourier analysis and applications. Linear Algebra Appl. 419, 180–233 (2006)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Serra-Capizzano, S., Tablino Possio, C.: Spectral and structural analysis of high precision finite difference matrices for elliptic operators. Linear Algebra Appl. 293, 85–131 (1999)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Serra-Capizzano, S., Tablino Possio, C.: Analysis of preconditioning strategies for collocation linear systems. Linear Algebra Appl. 369, 41–75 (2003)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Tilli, P.: Locally Toeplitz sequences: spectral properties and applications. Linear Algebra Appl. 278, 91–120 (1998)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Tilli, P.: A note on the spectral distribution of Toeplitz matrices. Linear Multilinear Algebra 45, 147–159 (1998)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Vichnevetsky, R.: Wave propagation and reflection in irregular grids for hyperbolic equations. Appl. Numer. Math. 3, 133–166 (1987)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Zuazua, E.: Propagation, observation, and control of waves approximated by finite difference methods. SIAM Rev. 47, 197–243 (2005)MathSciNetCrossRefGoogle Scholar

Copyright information

© Istituto di Informatica e Telematica del Consiglio Nazionale delle Ricerche 2018

Authors and Affiliations

  1. 1.Department of Science and High TechnologyUniversity of InsubriaComoItaly
  2. 2.Department of Information TechnologyUppsala UniversityUppsalaSweden

Personalised recommendations