Negative norm estimates for fully discrete finite element approximations to the wave equation with nonhomogeneous 𝐿₂ Dirichlet boundary data

Bales, L.; Lasiecka, I.

doi:10.1090/S0025-5718-1995-1262280-9

Negative norm estimates for fully discrete finite element approximations to the wave equation with nonhomogeneous $L_ 2$ Dirichlet boundary data
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Math. Comp. 64 (1995), 89-115 Request permission

Abstract:

This paper treats time and space finite element approximations of the solution to the nonhomogeneous wave equation with ${L_2}$ boundary terms and smooth right-hand side. For the case of ${L_2}$ boundary data, the rates of convergence in negative norms are derived. In the case of smooth forcing term and zero boundary data, optimal rates of convergence in "positive" norms are provided.

References

A. K. Aziz and Peter Monk, Continuous finite elements in space and time for the heat equation, Math. Comp. 52 (1989), no. 186, 255–274. MR 983310, DOI 10.1090/S0025-5718-1989-0983310-2
Garth A. Baker and James H. Bramble, Semidiscrete and single step fully discrete approximations for second order hyperbolic equations, RAIRO Anal. Numér. 13 (1979), no. 2, 75–100 (English, with French summary). MR 533876, DOI 10.1051/m2an/1979130200751
Laurence A. Bales, Semidiscrete and single step fully discrete approximations for second order hyperbolic equations with time-dependent coefficients, Math. Comp. 43 (1984), no. 168, 383–414. MR 758190, DOI 10.1090/S0025-5718-1984-0758190-6
L. A. Bales, Semidiscrete and single step fully discrete finite element approximations for second order hyperbolic equations with nonsmooth solutions, RAIRO Modél. Math. Anal. Numér. 27 (1993), no. 1, 55–63 (English, with English and French summaries). MR 1204628, DOI 10.1051/m2an/1993270100551
Laurence A. Bales, Some remarks on post-processing and negative norm estimates for approximations to nonsmooth solutions of hyperbolic equations, Comm. Numer. Methods Engrg. 9 (1993), no. 8, 701–710. MR 1237248, DOI 10.1002/cnm.1640090809
L. Bales and I. Lasiecka, Continuous finite elements in space and time for the nonhomogeneous wave equation, Comput. Math. Appl. 27 (1994), no. 3, 91–102. MR 1255008, DOI 10.1016/0898-1221(94)90048-5
Alain Bensoussan, Giuseppe Da Prato, Michel C. Delfour, and Sanjoy K. Mitter, Representation and control of infinite-dimensional systems. Vol. II, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1246331, DOI 10.1007/978-1-4612-2750-2
J. H. Bramble and A. H. Schatz, Higher order local accuracy by averaging in the finite element method, Math. Comp. 31 (1977), no. 137, 94–111. MR 431744, DOI 10.1090/S0025-5718-1977-0431744-9
Gilbert Choudury and Irena Lasiecka, Optimal convergence rates for semidiscrete approximations of parabolic problems with nonsmooth boundary data, Numer. Funct. Anal. Optim. 12 (1991), no. 5-6, 469–485 (1992). MR 1159921, DOI 10.1080/01630569108816443

A space-time finite element method for the second order wave equation

P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 775683
Pierre Grisvard, Exact controllability of the wave equation in presence of corners and cracks, Semigroup theory and evolution equations (Delft, 1989) Lecture Notes in Pure and Appl. Math., vol. 135, Dekker, New York, 1991, pp. 213–225. MR 1164652
Kenneth Hoffman and Ray Kunze, Linear algebra, 2nd ed., Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971. MR 0276251
Bernie L. Hulme, One-step piecewise polynomial Galerkin methods for initial value problems, Math. Comp. 26 (1972), 415–426. MR 321301, DOI 10.1090/S0025-5718-1972-0321301-2
Claes Johnson and Uno Nävert, An analysis of some finite element methods for advection-diffusion problems, Analytical and numerical approaches to asymptotic problems in analysis (Proc. Conf., Univ. Nijmegen, Nijmegen, 1980) North-Holland Math. Stud., vol. 47, North-Holland, Amsterdam-New York, 1981, pp. 99–116. MR 605502
I. Lasiecka and J. Sokołowski, Semidiscrete approximations of hyperbolic boundary value problems with nonhomogeneous Dirichlet boundary conditions, SIAM J. Math. Anal. 20 (1989), no. 6, 1366–1387. MR 1019306, DOI 10.1137/0520090
I. Lasiecka, J.-L. Lions, and R. Triggiani, Nonhomogeneous boundary value problems for second order hyperbolic operators, J. Math. Pures Appl. (9) 65 (1986), no. 2, 149–192. MR 867669
Michael S. Mock and Peter D. Lax, The computation of discontinuous solutions of linear hyperbolic equations, Comm. Pure Appl. Math. 31 (1978), no. 4, 423–430. MR 468216, DOI 10.1002/cpa.3160310403
J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 2, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 9, Masson, Paris, 1988 (French). Perturbations. [Perturbations]. MR 963060
J.-L. Lions, Espaces d’interpolation et domaines de puissances fractionnaires d’opérateurs, J. Math. Soc. Japan 14 (1962), 233–241 (French). MR 152878, DOI 10.2969/jmsj/01420233

Nonhomogeneous boundary value problems and applications

Semi-groups of linear operators and applications to partial differential equations

Vidar Thomée, Galerkin finite element methods for parabolic problems, Lecture Notes in Mathematics, vol. 1054, Springer-Verlag, Berlin, 1984. MR 744045
Ragnar Winther, A stable finite element method for initial-boundary value problems for first-order hyperbolic systems, Math. Comp. 36 (1981), no. 153, 65–86. MR 595042, DOI 10.1090/S0025-5718-1981-0595042-6