Abstract
In this chapter, we briefly present the conclusions of this book and we list some open problems related to the issues discussed in this monograph.
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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)
Ainsworth M.: Dispersive and dissipative behaviour of high order DG finite element methods. J. Comput. Phys. 198(1), 106–130 (2004)
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)
Antonietti P.F., Buffa A., Perugia I.: DG approximation of the Laplace eigenproblem. Comput. Methods Appl. Mech. Engrg. 195(25–28), 3483–3505 (2006)
Arnold D.N.: An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19, 742–760 (1982)
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)
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)
Bahouri H., Chemin J.Y., Danchin R.: Fourier Analysis and Nonlinear PDEs. Springer, New York (2011)
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)
Belytschko T., Mullen R.: On dispersive properties of finite element solutions. Modern Problems in Elastic Wave Propagation. Wiley, New York (1978)
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)
Brillouin L.: Wave Propagation and Group Velocity. Academic Press, New York (1960)
Burq N., Gérard P.: Contrôle optimal des equations aux derivées partielles. Cours de l’École Polytechnique (2002)
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)
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)
Chironis N.P., Sclater N.: Mechanisms and Mechanical Devices Sourcebook, 3th edn. McGraw-Hill, New York (2001)
Cockburn B.: Discontinuous Galerkin methods. Z. Agnew. Math. Mech. 83(11), 731–754 (2003)
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)
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)
Cockburn B., Shu C.W.: The local DG finite element method for convection-diffusion systems. SIAM J. Numer. Anal. 35, 2440–2463 (1998)
Dacorogna B.: Direct Methods in the Calculus of Variations. Springer, Berlin (1989)
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)
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)
Ervedoza S., Marica A., Zuazua E.: Uniform observability property for discrete waves on strictly concave non-uniform meshes: a multiplier approach, in preparation.
Ervedoza S., Zheng C., Zuazua E.: On the observability of time-discrete conservative linear systems. J. Funct. Anal. 254(12), 3037–3078 (2008)
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)
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)
Ervedoza S., Zuazua E.: On the numerical approximation of controls for waves. Springer Briefs in Mathematics. Springer, New York (2013)
Feng X., Wu H.: DG methods for the Helmholtz equation with large wave number. SIAM J. Numer. Anal. 47(4), 2872–2896 (2009)
Gérard P., Markowich P.A., Mauser N.J., Poupaud F.: Homogenization limits and Wigner transforms. Comm. Pure Appl. Math. (50)(4), 323–379 (1997)
Glowinski R.: Ensuring well-posedness by analogy: Stokes problem and boundary control for the wave equation. J. Comput. Phys. 103(2), 189–221 (1992)
Glowinski R., He J., Lions J.L.: Exact and approximate controllability for distributed parameter systems: a numerical approach. Cambridge University Press, Cambridge (2008)
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)
Grenander U., Szegö G.: Toeplitz Forms and Their Applications. Chelsea Publishing, New York (1984)
Grote M.J., Schneebeli A., Schötzau D.: DG finite element method for the wave equation. SIAM J. Numer. Anal. 44(6), 2408–2431 (2006)
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)
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)
Ignat L., Zuazua E.: Numerical dispersive schemes for the nonlinear Schrödinger equation. SIAM J. Numer. Anal. 47(2), 1366–1390 (2009)
Infante J.A., Zuazua E.: Boundary observability for the space-discretizations of the one-dimensional wave equation. M2AN, 33(2), 407–438 (1999)
Kalman R.E.: On the general theory of control systems, Proceeding of First International Congress of IFAC, vol. 1, pp. 481–492. Moscow (1960)
Lebeau G.:Contrôle de l’équation de Schrödinger. J. Math. Pures Appl. 71, 267–291 (1992)
Lieb E.H., Loss M.: Analysis. Graduate Studies in Mathematics, vol. 14. AMS, Providence, RI (1997)
Linares F., Ponce G.: Introduction to Nonlinear Dispersive Equations. Springer, New York (2009)
Lions J.L.: Contrôlabilité Exacte, Perturbations et Stabilisation des Systèmes Distribués, vol. 1. Masson, Paris (1988)
Loreti P., Mehrenberger M.: An Ingham type proof for a two-grid observability theorem. ESAIM: COCV 14(3), 604–631 (2008)
Macià F.: Propagación y control de vibraciones en medios discretos y continuos. PhD Thesis, Universidad Complutense de Madrid (2002)
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)
Markowich P.A., Poupaud F.: The Pseudo-Differential Approach to Finite Difference Revisited, vol. 36, pp. 161–186. Calcolo, Springer, New York (1999)
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)
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)
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)
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)
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)
Marica A., Zuazua E.: Propagation of 1 − d waves in regular discrete heterogeneous media: a Wigner measure approach, accepted in FoCM
Mariegaard J.S.: Numerical approximation of boundary control for the wave equation. Ph.D. Thesis, Department of Mathematics, Technical University of Denmark (2009)
Micu S.: Uniform boundary controllability of a semi-discrete 1 − d wave equation. Numer. Math. 91(4), 723–768 (2002)
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)
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)
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)
Rose H.E.: Linear Algebra. A pure Mathematical Approach. Springer, New York (2002)
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)
Showalter R.E.: Monotone operators in Banach space and nonlinear partial differential equations. Mathematical Surveys and Monographs, vol. 49. AMS, Providence, RI (1997)
Simon J.: Compact sets in the space L p(0, T; B). Annali di matematica pura ed applicata IV(CXLVI), 65–96 (1987)
Sontag E.: Mathematical Control Theory: Deterministic Finite Dimensional Systems. Springer, New York (1998)
Stein E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press (1970)
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)
Trefethen L.N.: Group velocity in finite difference schemes. SIAM Rev. 24(2), 113–136 (1982)
Zuazua E.: Exponential decay for the semilinear wave equation with localized damping in unbounded domains. J. Math. Pures et Appl. 70, 513–529 (1991)
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)
Zuazua E.: Propagation, observation, control and numerical approximation of waves. SIAM Review 47(2), 197–243 (2005)
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)
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Marica, A., Zuazua, E. (2014). Comments and Open Problems. In: Symmetric Discontinuous Galerkin Methods for 1-D Waves. SpringerBriefs in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-5811-1_8
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