Filtering Mechanisms

  • Aurora Marica
  • Enrique Zuazua
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)


In this chapter, we design several filtering strategies for the discontinuous Galerkin approximations of the wave and Klein–Gordon equations based on the classical Fourier truncation method or on the bi-grid filtering algorithm. We rigorously prove their efficiency to recover the uniform observability estimates as the mesh size parameter goes to zero. In the last section of this chapter, we present several numerical simulations showing in particular how solutions corresponding to bi-grid projections of highly oscillatory Gaussian profiles split into several wave packets.


Initial Data Wave Packet Gordon Equation Physical Mode Fourier Representation 
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  1. 1.
    Agut C., Diaz J.: Stability analysis of the interior penalty DG method for the wave equation. ESAIM: Math. Model. Numer. Anal. 47(3), 903–932 (2013)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Ainsworth M.: Dispersive and dissipative behaviour of high order DG finite element methods. J. Comput. Phys. 198(1), 106–130 (2004)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Ainsworth M., Monk P., Muniz W.: Dispersive and dissipative properties of DG finite element methods for the second-order wave equation. J. Sci. Comput. 27(1–3), 5–40 (2006)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Antonietti P.F., Buffa A., Perugia I.: DG approximation of the Laplace eigenproblem. Comput. Methods Appl. Mech. Engrg. 195(25–28), 3483–3505 (2006)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Arnold D.N.: An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19, 742–760 (1982)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Arnold D.N., Brezzi F., Cockburn B., Marini L.D.: Unified analysis of DG methods for elliptic problems. SIAM J. Numer. Anal. 39, 1749–1779 (2002)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Atkins H.L., Hu F.Q.: Eigensolution analysis of the DG method with nonuniform grids: I. One space dimension. J. Comput. Phys. 182(2), 516–545 (2002)MATHMathSciNetGoogle Scholar
  8. 8.
    Bahouri H., Chemin J.Y., Danchin R.: Fourier Analysis and Nonlinear PDEs. Springer, New York (2011)Google Scholar
  9. 9.
    Bardos C., Lebeau G., Rauch J.: Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary. SIAM J. Contr. Optim. 30, 1024–1065 (1992)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Belytschko T., Mullen R.: On dispersive properties of finite element solutions. Modern Problems in Elastic Wave Propagation. Wiley, New York (1978)Google Scholar
  11. 11.
    Brezzi F., Cockburn B., Marini L.D., Süli E.: Stabilization mechanisms in DG finite element methods. J. Comput. Methods Appl. Mech. Engrg. 195, 3293–3310 (2006)CrossRefMATHGoogle Scholar
  12. 12.
    Brillouin L.: Wave Propagation and Group Velocity. Academic Press, New York (1960)MATHGoogle Scholar
  13. 13.
    Burq N., Gérard P.: Contrôle optimal des equations aux derivées partielles. Cours de l’École Polytechnique (2002)Google Scholar
  14. 14.
    Calo V., Demkowicz L., Gopalakrishnan J., Muga I., Pardo D., Zitelli J.: A class of discontinuous Petrov-Galerkin methods. Part IV: The optimal test norm and time-harmonic wave propagation in 1 − d. J. Comput. Phys. 230, 2406–2432 (2011)Google Scholar
  15. 15.
    Castro C., Micu S.: Boundary controllability of a linear semi-discrete 1 − d wave equation derived from a mixed finite element method. Numer. Math. 102(3), 413–462 (2006)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Chironis N.P., Sclater N.: Mechanisms and Mechanical Devices Sourcebook, 3th edn. McGraw-Hill, New York (2001)Google Scholar
  17. 17.
    Cockburn B.: Discontinuous Galerkin methods. Z. Agnew. Math. Mech. 83(11), 731–754 (2003)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Cockburn B., Gopalakrishnan J., Lazarov R.: Unified hybridization of DG, mixed and continuous Galerkin methods for second-order elliptic problems. SIAM J. Numer. Anal. 47(2), 1319–1365 (2009)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Cockburn B., Karniadakis G., Shu C.W.: The development of DG methods, in Discontinuous Galerkin methods. Theory, Computation and Applications, Lecture Notes in Computational Science and Engineering, vol. 11, pp. 3–50. Springer, Berlin (2000)Google Scholar
  20. 20.
    Cockburn B., Shu C.W.: The local DG finite element method for convection-diffusion systems. SIAM J. Numer. Anal. 35, 2440–2463 (1998)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Dacorogna B.: Direct Methods in the Calculus of Variations. Springer, Berlin (1989)CrossRefMATHGoogle Scholar
  22. 22.
    Demkowicz L., Gopalakrishnan J., Muga I., Zitelli J.: Wavenumber explicit analysis of a DPG method for the multidimensional Helmholtz equation. Comput. Meth. Appl. Mechan. Eng. 213–216 (2012)Google Scholar
  23. 23.
    Ervedoza S.: Observability properties of a semi-discrete 1 − d wave equation derived from a mixed finite element method on non-uniform meshes. ESAIM: COCV 16(2), 298–326 (2010)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Ervedoza S., Marica A., Zuazua E.: Uniform observability property for discrete waves on strictly concave non-uniform meshes: a multiplier approach, in preparation.Google Scholar
  25. 25.
    Ervedoza S., Zheng C., Zuazua E.: On the observability of time-discrete conservative linear systems. J. Funct. Anal. 254(12), 3037–3078 (2008)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Ervedoza S., Zuazua E.: A systematic method for building smooth controls for smooth data. Discrete Contin. Dyn. Syst. Ser. B 14(4), 1375–1401 (2010)CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Ervedoza S., Zuazua E.: The wave equation: control and numerics, in Control of PDEs. Cetraro, Italy 2010, In: Cannarsa P.M and Coron J.M. (eds.), Lecture Notes in Mathematics, vol. 2048, pp. 245–339. Springer, New York (2012)Google Scholar
  28. 28.
    Ervedoza S., Zuazua E.: On the numerical approximation of controls for waves. Springer Briefs in Mathematics. Springer, New York (2013)CrossRefGoogle Scholar
  29. 29.
    Feng X., Wu H.: DG methods for the Helmholtz equation with large wave number. SIAM J. Numer. Anal. 47(4), 2872–2896 (2009)CrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    Gérard P., Markowich P.A., Mauser N.J., Poupaud F.: Homogenization limits and Wigner transforms. Comm. Pure Appl. Math. (50)(4), 323–379 (1997)Google Scholar
  31. 31.
    Glowinski R.: Ensuring well-posedness by analogy: Stokes problem and boundary control for the wave equation. J. Comput. Phys. 103(2), 189–221 (1992)CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    Glowinski R., He J., Lions J.L.: Exact and approximate controllability for distributed parameter systems: a numerical approach. Cambridge University Press, Cambridge (2008)CrossRefGoogle Scholar
  33. 33.
    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. Japan J. Appl. Math. 7(1), 1–76 (1990)MATHMathSciNetGoogle Scholar
  34. 34.
    Grenander U., Szegö G.: Toeplitz Forms and Their Applications. Chelsea Publishing, New York (1984)MATHGoogle Scholar
  35. 35.
    Grote M.J., Schneebeli A., Schötzau D.: DG finite element method for the wave equation. SIAM J. Numer. Anal. 44(6), 2408–2431 (2006)CrossRefMATHMathSciNetGoogle Scholar
  36. 36.
    Hu F.Q., Hussaini M.Y., Rasetarinera P.: An analysis of the DG method for wave propagation problems. J. Comput. Phys. 151, 921–946 (1999)CrossRefMATHGoogle Scholar
  37. 37.
    Ignat L., Zuazua E.: Convergence of a two-grid algorithm for the control of the wave equation. J. Eur. Math. Soc. 11(2), 351–391 (2009)CrossRefMATHMathSciNetGoogle Scholar
  38. 38.
    Ignat L., Zuazua E.: Numerical dispersive schemes for the nonlinear Schrödinger equation. SIAM J. Numer. Anal. 47(2), 1366–1390 (2009)CrossRefMATHMathSciNetGoogle Scholar
  39. 39.
    Infante J.A., Zuazua E.: Boundary observability for the space-discretizations of the one-dimensional wave equation. M2AN, 33(2), 407–438 (1999)Google Scholar
  40. 40.
    Kalman R.E.: On the general theory of control systems, Proceeding of First International Congress of IFAC, vol. 1, pp. 481–492. Moscow (1960)Google Scholar
  41. 41.
    Lebeau G.:Contrôle de l’équation de Schrödinger. J. Math. Pures Appl. 71, 267–291 (1992)Google Scholar
  42. 42.
    Lieb E.H., Loss M.: Analysis. Graduate Studies in Mathematics, vol. 14. AMS, Providence, RI (1997)Google Scholar
  43. 43.
    Linares F., Ponce G.: Introduction to Nonlinear Dispersive Equations. Springer, New York (2009)MATHGoogle Scholar
  44. 44.
    Lions J.L.: Contrôlabilité Exacte, Perturbations et Stabilisation des Systèmes Distribués, vol. 1. Masson, Paris (1988)Google Scholar
  45. 45.
    Loreti P., Mehrenberger M.: An Ingham type proof for a two-grid observability theorem. ESAIM: COCV 14(3), 604–631 (2008)CrossRefMATHMathSciNetGoogle Scholar
  46. 46.
    Macià F.: Propagación y control de vibraciones en medios discretos y continuos. PhD Thesis, Universidad Complutense de Madrid (2002)Google Scholar
  47. 47.
    Macìa F.: Wigner measures in the discrete setting: high frequency analysis of sampling and reconstruction operators. SIAM J. Math. Anal. 36(2) 347–383 (2004)CrossRefMATHMathSciNetGoogle Scholar
  48. 48.
    Markowich P.A., Poupaud F.: The Pseudo-Differential Approach to Finite Difference Revisited, vol. 36, pp. 161–186. Calcolo, Springer, New York (1999)Google Scholar
  49. 49.
    Marica A., Zuazua E.: Localized solutions for the finite difference semi-discretization of the wave equation. C. R. Acad. Sci. Paris Ser. I 348, 647–652 (2010)CrossRefMATHMathSciNetGoogle Scholar
  50. 50.
    Marica A., Zuazua E.: Localized solutions and filtering mechanisms for the DG semi-discretizations of the 1 − d wave equation. C. R. Acad. Sci. Paris Ser. I 348, 1087–1092 (2010)CrossRefMATHMathSciNetGoogle Scholar
  51. 51.
    Marica A., Zuazua E.: High frequency wave packets for the Schrödinger equation and its numerical approximations. C. R. Acad. Sci. Paris Ser. I 349, 105–110 (2011)CrossRefMATHMathSciNetGoogle Scholar
  52. 52.
    Marica A., Zuazua E.: On the quadratic finite element approximation of one-dimensional waves: propagation, observation, and control. SIAM J. Numer. Anal. 50(5), 2744–2777 (2012)CrossRefMATHMathSciNetGoogle Scholar
  53. 53.
    Marica A., Zuazua E.: On the quadratic finite element approximation of 1 − d waves: propagation, observation, control and numerical implementation, CFL-80: A Celebration of 80 Years of the Discovery of CFL Condition. Kubrusly, C., Moura C.A. (eds.) Springer Proceedings in Mathematics, pp. 75–100 Springer, New York (2012)Google Scholar
  54. 54.
    Marica A., Zuazua E.: Propagation of 1 − d waves in regular discrete heterogeneous media: a Wigner measure approach, accepted in FoCMGoogle Scholar
  55. 55.
    Mariegaard J.S.: Numerical approximation of boundary control for the wave equation. Ph.D. Thesis, Department of Mathematics, Technical University of Denmark (2009)Google Scholar
  56. 56.
    Micu S.: Uniform boundary controllability of a semi-discrete 1 − d wave equation. Numer. Math. 91(4), 723–768 (2002)CrossRefMATHMathSciNetGoogle Scholar
  57. 57.
    Micu S.: Uniform boundary controllability of a semi-discrete 1 − d wave equation with vanishing viscosity. SIAM J. Control Optim. 47(6), 2857–2885 (2008)CrossRefMATHMathSciNetGoogle Scholar
  58. 58.
    Micu S., Zuazua E.: An Introduction to the controllability of partial differential equations, in Quelques questions de théorie du contrôle. In: Sari T. (ed.) Collection Travaux en Cours, pp. 69–157. Hermann, Paris (2005)Google Scholar
  59. 59.
    Negreanu M., Zuazua E.: Convergence of a multigrid method for the controllability of a 1 − d wave equation. C. R. Math. Acad. Sci. Paris 338, 413–418 (2004)CrossRefMATHMathSciNetGoogle Scholar
  60. 60.
    Rose H.E.: Linear Algebra. A pure Mathematical Approach. Springer, New York (2002)CrossRefMATHGoogle Scholar
  61. 61.
    Sherwin S.:Dispersion analysis of the continuous and DG formulations, in Discontinuous Galerkin methods. Theory, Computation and Applications, Lecture Notes in Computational Science and Engineering, vol. 11, pp. 425–431. Springer, New York (2000)Google Scholar
  62. 62.
    Showalter R.E.: Monotone operators in Banach space and nonlinear partial differential equations. Mathematical Surveys and Monographs, vol. 49. AMS, Providence, RI (1997)Google Scholar
  63. 63.
    Simon J.: Compact sets in the space L p(0, T; B). Annali di matematica pura ed applicata IV(CXLVI), 65–96 (1987)Google Scholar
  64. 64.
    Sontag E.: Mathematical Control Theory: Deterministic Finite Dimensional Systems. Springer, New York (1998)CrossRefMATHGoogle Scholar
  65. 65.
    Stein E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press (1970)Google Scholar
  66. 66.
    Tebou Tcheougoué L.R., Zuazua E.: Uniform boundary stabilization of the finite difference space discretization of the 1 − d wave equation. Adv. Comput. Math. 26(1–3), 337–365 (2007)CrossRefMathSciNetGoogle Scholar
  67. 67.
    Trefethen L.N.: Group velocity in finite difference schemes. SIAM Rev. 24(2), 113–136 (1982)CrossRefMATHMathSciNetGoogle Scholar
  68. 68.
    Zuazua E.: Exponential decay for the semilinear wave equation with localized damping in unbounded domains. J. Math. Pures et Appl. 70, 513–529 (1991)MATHMathSciNetGoogle Scholar
  69. 69.
    Zuazua E.: Boundary observability for the finite difference space semi-discretizations of the 2 − d wave equation in the square. J. Math. Pures Appl. 78(5) 523–563 (1999)CrossRefMATHMathSciNetGoogle Scholar
  70. 70.
    Zuazua E.: Propagation, observation, control and numerical approximation of waves. SIAM Review 47(2), 197–243 (2005)CrossRefMATHMathSciNetGoogle Scholar
  71. 71.
    Zuazua E.: Controllability and observability of PDEs: Some results and open problems. In: Handbook of Differential Equations: Evolutionary Equations, vol. 3, pp. 527–621. In Dafermos C. M. and Feireisl E. (eds.) Elsevier B. V. North Holland (2007)Google Scholar

Copyright information

© Aurora Marica, Enrique Zuazua 2014

Authors and Affiliations

  • Aurora Marica
    • 1
  • Enrique Zuazua
    • 2
  1. 1.Institute of Mathematics and Scientific ComputingUniversity of GrazGrazAustria
  2. 2.BCAM-Basque Center for Applied Mathematics IkerbasqueBilbaoSpain

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