Approximations for weakly nonlinear evolution equations
HTML articles powered by AMS MathViewer
- by Milan Miklavčič PDF
- Math. Comp. 53 (1989), 471-484 Request permission
Abstract:
Convergence of approximations for a large class of weakly nonlinear parabolic and hyperbolic equations is proven. The main emphasis is on proving convergence of finite element and spectral Galerkin approximations of solutions to the weakly nonlinear wave equation \[ u”(t) + Au(t) = F(t,u(t),u’(t)),\quad u(0) = {x_0},u’(0) = {y_0},\] under minimal assumptions on the linear operator A and on the approximation spaces. A can be a very general elliptic operator (not just of 2nd order and not necessarily in a bounded domain); A can also be very singular and degenerate. The results apply also to systems of equations. Verification of the hypotheses is completely elementary for a large class of problems.References
- Shmuel Agmon, The coerciveness problem for integro-differential forms, J. Analyse Math. 6 (1958), 183–223. MR 132912, DOI 10.1007/BF02790236
- J. M. Ball, Strongly continuous semigroups, weak solutions, and the variation of constants formula, Proc. Amer. Math. Soc. 63 (1977), no. 2, 370–373. MR 442748, DOI 10.1090/S0002-9939-1977-0442748-6
- H. T. Banks and K. Kunisch, An approximation theory for nonlinear partial differential equations with applications to identification and control, SIAM J. Control Optim. 20 (1982), no. 6, 815–849. MR 675572, DOI 10.1137/0320059
- Philip Brenner and Vidar Thomée, On rational approximations of semigroups, SIAM J. Numer. Anal. 16 (1979), no. 4, 683–694. MR 537280, DOI 10.1137/0716051
- C. Canuto and A. Quarteroni, Approximation results for orthogonal polynomials in Sobolev spaces, Math. Comp. 38 (1982), no. 157, 67–86. MR 637287, DOI 10.1090/S0025-5718-1982-0637287-3 S. N. Chow, D. R. Dunninger & M. Miklavčič, "Galerkin approximations for singular elliptic and semilinear parabolic problems," SIAM J. Numer. Anal., submitted.
- Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 0520174
- Roland Glowinski, Numerical methods for nonlinear variational problems, Springer Series in Computational Physics, Springer-Verlag, New York, 1984. MR 737005, DOI 10.1007/978-3-662-12613-4
- Roland Glowinski, Jacques-Louis Lions, and Raymond Trémolières, Numerical analysis of variational inequalities, Studies in Mathematics and its Applications, vol. 8, North-Holland Publishing Co., Amsterdam-New York, 1981. Translated from the French. MR 635927
- Jerome A. Goldstein, Semigroups of linear operators and applications, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1985. MR 790497
- Jack K. Hale, Asymptotic behavior of dissipative systems, Mathematical Surveys and Monographs, vol. 25, American Mathematical Society, Providence, RI, 1988. MR 941371, DOI 10.1090/surv/025
- Jack K. Hale, Xiao-Biao Lin, and Geneviève Raugel, Upper semicontinuity of attractors for approximations of semigroups and partial differential equations, Math. Comp. 50 (1988), no. 181, 89–123. MR 917820, DOI 10.1090/S0025-5718-1988-0917820-X
- Hans-Peter Helfrich, Error estimates for semidiscrete Galerkin type approximations to semilinear evolution equations with nonsmooth initial data, Numer. Math. 51 (1987), no. 5, 559–569. MR 910865, DOI 10.1007/BF01400356
- Daniel Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. MR 610244, DOI 10.1007/BFb0089647
- Reuben Hersh and Tosio Kato, High-accuracy stable difference schemes for well-posed initial value problems, SIAM J. Numer. Anal. 16 (1979), no. 4, 670–682. MR 537279, DOI 10.1137/0716050
- Claes Johnson, Stig Larsson, Vidar Thomée, and Lars B. Wahlbin, Error estimates for spatially discrete approximations of semilinear parabolic equations with nonsmooth initial data, Math. Comp. 49 (1987), no. 180, 331–357. MR 906175, DOI 10.1090/S0025-5718-1987-0906175-1 T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Springer, New York, 1980.
- Tosio Kato, Remarks on pseudo-resolvents and infinitesimal generators of semi-groups, Proc. Japan Acad. 35 (1959), 467–468. MR 117570
- Thomas G. Kurtz, Extensions of Trotter’s operator semigroup approximation theorems, J. Functional Analysis 3 (1969), 354–375. MR 0242016, DOI 10.1016/0022-1236(69)90031-7
- Patricia K. Lamm and Katherine A. Murphy, Estimation of discontinuous coefficients and boundary parameters for hyperbolic systems, Quart. Appl. Math. 46 (1988), no. 1, 1–22. MR 934677, DOI 10.1090/S0033-569X-1988-0934677-1 J. L. Lions & E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Springer, New York, 1972.
- E. Magenes, R. H. Nochetto, and C. Verdi, Energy error estimates for a linear scheme to approximate nonlinear parabolic problems, RAIRO Modél. Math. Anal. Numér. 21 (1987), no. 4, 655–678 (English, with French summary). MR 921832, DOI 10.1051/m2an/1987210406551
- Milan Miklavčič, Stability for semilinear parabolic equations with noninvertible linear operator, Pacific J. Math. 118 (1985), no. 1, 199–214. MR 783024, DOI 10.2140/pjm.1985.118.199 M. Miklavčič, "Galerkin approximations of semilinear parabolic equations," preprint, 1986.
- A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486, DOI 10.1007/978-1-4612-5561-1
- Martin Schechter, On the invariance of the essential spectrum of an arbitrary operator. II, Ricerche Mat. 16 (1967), 3–26. MR 230153
- Irving Segal, Non-linear semi-groups, Ann. of Math. (2) 78 (1963), 339–364. MR 152908, DOI 10.2307/1970347 V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Lecture Notes in Math., vol. 1054, Springer, New York, 1984.
- H. F. Trotter, Approximation of semi-groups of operators, Pacific J. Math. 8 (1958), 887–919. MR 103420, DOI 10.2140/pjm.1958.8.887
- Teruo Ushijima, Approximation theory for semi-groups of linear operators and its application to approximation of wave equations, Japan. J. Math. (N.S.) 1 (1975/76), no. 1, 185–224. MR 420340, DOI 10.4099/math1924.1.185
- Kôsaku Yosida, Functional analysis, 6th ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 123, Springer-Verlag, Berlin-New York, 1980. MR 617913
Additional Information
- © Copyright 1989 American Mathematical Society
- Journal: Math. Comp. 53 (1989), 471-484
- MSC: Primary 65J15; Secondary 65Mxx
- DOI: https://doi.org/10.1090/S0025-5718-1989-0982370-2
- MathSciNet review: 982370