Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Some remarks on the convergence of approximate solutions of nonlinear evolution equations in Hilbert spaces

Author: Laurent Véron
Journal: Math. Comp. 39 (1982), 325-337
MSC: Primary 47H15; Secondary 34A45
MathSciNet review: 669633
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \partial \Phi $ be the subdifferential of some lower semicontinuous convex function $ \Phi $ of a real Hilbert space H, $ f \in {L^2}(0,T;H)$ and $ {u_n}$ a continouous piecewise linear approximate solution of $ du/dt + \partial \Phi (u) \ni f$, obtained by an implicit scheme. If $ {u_0} \in \operatorname{Dom} (\Phi )$, then $ d{u_n}/dt$ converges to $ du/dt$ in $ {L^2}(0,T;H)$. Moreover, if $ {u_0} \in \overline {\operatorname{Dom} (\partial \Phi )} $, we construct a step function $ {\eta _n}(t)$ approximating t such that $ {\lim _{n \to + \infty }}\smallint _0^T{\eta _n}\vert d{u_n}/dt - du/dt{\vert^2}\;dt = 0$. When $ \Phi $ is inf-compact and when the sequence of approximation of f is weakly convergent to f, then $ {u_n}$ converges to u in $ C([0,T];H)$ and $ {\eta _n}d{u_n}/dt$ is weakly convergent to $ tdu/dt$.

References [Enhancements On Off] (What's this?)

  • [1] Hédy Attouch, Convergence de fonctionnelles convexes, Journées d’Analyse Non Linéaire (Proc. Conf., Besançon, 1977), Lecture Notes in Math., vol. 665, Springer, Berlin, 1978, pp. 1–40 (French). MR 519420
  • [2] Pierre Baras, Compacité de l’opérateur 𝑓\mapsto𝑢 solution d’une équation non linéaire (𝑑𝑢/𝑑𝑡)+𝐴𝑢∋𝑓, C. R. Acad. Sci. Paris Sér. A-B 286 (1978), no. 23, A1113–A1116 (French, with English summary). MR 493554
  • [3] Ph. Benilan, Equations d'Evolution dans un Espace de Banach Quelconque et Applications, Thèse, Université de Paris XI, Orsay, Paris, 1972.
  • [4] H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973 (French). North-Holland Mathematics Studies, No. 5. Notas de Matemática (50). MR 0348562
  • [5] Haïm Brézis, Asymptotic behavior of some evolution systems, Nonlinear evolution equations (Proc. Sympos., Univ. Wisconsin, Madison, Wis., 1977) Publ. Math. Res. Center Univ. Wisconsin, vol. 40, Academic Press, New York-London, 1978, pp. 141–154. MR 513816
  • [6] H. Brezis, Propriétés régularisantes de certains semi-groupes non linéaires, Israel J. Math. 9 (1971), 513–534 (French). MR 283635,
  • [7] Michael G. Crandall and L. C. Evans, On the relation of the operator ∂/∂𝑠+∂/∂𝜏 to evolution governed by accretive operators, Israel J. Math. 21 (1975), no. 4, 261–278. MR 390853,
  • [8] M. G. Crandall and T. M. Liggett, Generation of semi-groups of nonlinear transformations on general Banach spaces, Amer. J. Math. 93 (1971), 265–298. MR 287357,

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 47H15, 34A45

Retrieve articles in all journals with MSC: 47H15, 34A45

Additional Information

Article copyright: © Copyright 1982 American Mathematical Society