Skip to Main Content

Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

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

The 2024 MCQ for Mathematics of Computation is 1.78.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

On the convergence of difference approximations to nonlinear contraction semigroups in Hilbert spaces
HTML articles powered by AMS MathViewer

by Olavi Nevanlinna PDF
Math. Comp. 32 (1978), 321-334 Request permission

Abstract:

Convergence properties of the difference schemes (S) \[ {h^{ - 1}}\sum \limits _{j = 0}^k {{\alpha _j}{u_{n + j}}} + \sum \limits _{j = 0}^k {{\beta _j}A{u_{n + j}}} = 0,\quad n \geqslant 0,\], for evolution equations (E) \[ \frac {{du(t)}}{{dt}} + Au(t) = 0,\quad t \geqslant 0;\quad u(0) = {u_0} \in \overline {D(A)} \] are studied. Here A is a nonlinear, maximally monotone operator in a real Hilbert space. It is shown, in particular, that if the scheme (S) is consistent and stable for the test equation $x\prime = \lambda x$ for $\lambda \in {\text {C}} - K$, where K is a compact subset of the right half-plane, then (S) is convergent as $h \downarrow 0$, with suitable initial values, for (E), on compact intervals [0, T]. Moreover, the convergence is uniform on the half-axis $t \geqslant 0$, if the solution $u(t)$ tends strongly to a constant as $t \to \infty$. We also show that under weaker stability conditions one can construct conditionally convergent methods.
References
  • H. Brezis and A. Pazy, Semigroups of nonlinear contractions on convex sets, J. Functional Analysis 6 (1970), 237–281. MR 0448185, DOI 10.1016/0022-1236(70)90060-1
  • Ronald E. Bruck Jr., Asymptotic convergence of nonlinear contraction semigroups in Hilbert space, J. Functional Analysis 18 (1975), 15–26. MR 377609, DOI 10.1016/0022-1236(75)90027-0
  • 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, DOI 10.2307/2373376
  • Germund G. Dahlquist, A special stability problem for linear multistep methods, Nordisk Tidskr. Informationsbehandling (BIT) 3 (1963), 27–43. MR 170477, DOI 10.1007/bf01963532
  • Germund Dahlquist, Error analysis for a class of methods for stiff non-linear initial value problems, Numerical analysis (Proc. 6th Biennial Dundee Conf., Univ. Dundee, Dundee, 1975) Lecture Notes in Math., Vol. 506, Springer, Berlin, 1976, pp. 60–72. MR 0448898
  • G. DAHLQUIST, On the Relation of G-Stability to Other Stability Concepts for Linear Multistep Methods, Report TRITA-NA-7618, Dept. of Comput. Sci., Royal Inst. of Tech., 1976.
  • J. Kačur, The Rothe method and nonlinear parabolic equations of arbitrary order, Theory of nonlinear operators (Proc. Summer School, Neuchâtel, 1972) Schr. Zentralinst. Math. Mech. Akad. Wiss. DDR, Heft 20, Akademie-Verlag, Berlin, 1974, pp. 125–131. MR 0364880
  • Nobuyuki Kenmochi and Sinnosuke Oharu, Difference approximation of nonlinear evolution equations and semigroups of nonlinear operators, Publ. Res. Inst. Math. Sci. 10 (1974/75), no. 1, 147–207. MR 0388185, DOI 10.2977/prims/1195192177
  • Olavi Nevanlinna, On error bounds for $G$-stable methods, Nordisk Tidskr. Informationsbehandling (BIT) 16 (1976), no. 1, 79–84. MR 488767, DOI 10.1007/bf01940780
  • O. NEVANLINNA, On Multistep Methods for Nonlinear Initial Value Problems with an Application to Minimization of Convex Functionals, Report HTKK-MAT-A76, Inst. of Math., Helsinki Univ. of Tech., 1976.
  • Olavi Nevanlinna, On the numerical integration of nonlinear initial value problems by linear multistep methods, Nordisk Tidskr. Informationsbehandling (BIT) 17 (1977), no. 1, 58–71. MR 494953, DOI 10.1007/bf01932399
  • Tadayasu Takahashi, Convergence of difference approximation of nonlinear evolution equations and generation of semigroups, J. Math. Soc. Japan 28 (1976), no. 1, 96–113. MR 399978, DOI 10.2969/jmsj/02810096
Similar Articles
  • Retrieve articles in Mathematics of Computation with MSC: 47H15, 65J05
  • Retrieve articles in all journals with MSC: 47H15, 65J05
Additional Information
  • © Copyright 1978 American Mathematical Society
  • Journal: Math. Comp. 32 (1978), 321-334
  • MSC: Primary 47H15; Secondary 65J05
  • DOI: https://doi.org/10.1090/S0025-5718-1978-0513203-9
  • MathSciNet review: 0513203