On the convergence of difference approximations to nonlinear contraction semigroups in Hilbert spaces

Author:
Olavi Nevanlinna

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

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Abstract | References | Similar Articles | Additional Information

Abstract: Convergence properties of the difference schemes (S)

*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 for , where

*K*is a compact subset of the right half-plane, then (S) is convergent as , with suitable initial values, for (E), on compact intervals [0,

*T*]. Moreover, the convergence is uniform on the half-axis , if the solution tends strongly to a constant as . We also show that under weaker stability conditions one can construct conditionally convergent methods.

**[1]**H. BREZIS,*Opérateurs Maximaux Monotones*, North-Holland, Amsterdam, 1973. 2. H. BREZIS & A. PAZY, "Semi groups of nonlinear contractions on convex sets,"*J. Functional Analysis*, v. 6, 1970, pp. 237-281. MR**0448185 (56:6494)****[3]**R. BRUCK, "Asymptotic convergence of nonlinear contraction semi groups in Hilbert space,"*J. Functional Analysis*, v. 18, 1975, pp. 15-26. MR**0377609 (51:13780)****[4]**M. CRANDALL & T. LIGGETT, "Generation of semigroups of nonlinear transformations on general Banach spaces,"*Amer. J. Math.*, v. 93, 1971, pp. 265-298. MR**0287357 (44:4563)****[5]**G. DAHLQUIST, "A special stability problem for linear multistep methods,"*BIT*, v. 3, 1963, pp. 27-43. MR**0170477 (30:715)****[6]**G. DAHLQUIST,*Error Analysis for a Class of Methods for Stiff Non-Linear Initial Value Problems*, Lecture Notes in Math., vol. 506, Springer-Verlag, Berlin and New York, 1976, pp. 60-74. MR**0448898 (56:7203)****[7]**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.**[8]**J. KAČUR, "The Rothe method and nonlinear parabolic equations of arbitrary order,"*Theory of Nonlinear Operators*(Proc. Summer-school held in Oct. 1972 at Neuendorf), Akademie-Verlag, Berlin, 1974, pp. 125-131. MR**0364880 (51:1134)****[9]**N. KENMOCHI & S. OHARU, "Difference approximation of nonlinear evolution equations and semigroups of nonlinear operators,"*Publ. Res. Inst. Math. Sci.*, v. 10, 1974, pp. 147-207. MR**0388185 (52:9022)****[10]**O. NEVANLINNA, "On error bounds for*G*-stable methods,"*BIT*, v. 16, 1976, pp. 79-84. MR**0488767 (58:8281)****[11]**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.**[12]**O. NEVANLINNA, "On the numerical integration of nonlinear initial value problems by linear multistep methods,"*BIT*, v. 17, 1977, pp. 58-71. MR**0494953 (58:13728)****[13]**T. TAKAHASHI, "Convergence of difference approximation of nonlinear evolution equations and generation of semigroups,"*J. Math. Soc. Japan*, v. 28, 1976, pp. 96-113. MR**0399978 (53:3816)**

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DOI:
https://doi.org/10.1090/S0025-5718-1978-0513203-9

Article copyright:
© Copyright 1978
American Mathematical Society