On the convergence of difference approximations to nonlinear contraction semigroups in Hilbert spaces
Author:
Olavi Nevanlinna
Journal:
Math. Comp. 32 (1978), 321334
MSC:
Primary 47H15; Secondary 65J05
MathSciNet review:
0513203
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Convergence properties of the difference schemes (S) , for evolution equations (E) 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 for , where K is a compact subset of the right halfplane, then (S) is convergent as , with suitable initial values, for (E), on compact intervals [0, T]. Moreover, the convergence is uniform on the halfaxis , 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 and A.
Pazy, Semigroups of nonlinear contractions on convex sets, J.
Functional Analysis 6 (1970), 237–281. MR 0448185
(56 #6494)
 [3]
Ronald
E. Bruck Jr., Asymptotic convergence of nonlinear contraction
semigroups in Hilbert space, J. Funct. Anal. 18
(1975), 15–26. MR 0377609
(51 #13780)
 [4]
M.
G. Crandall and T.
M. Liggett, Generation of semigroups of nonlinear transformations
on general Banach spaces, Amer. J. Math. 93 (1971),
265–298. MR 0287357
(44 #4563)
 [5]
Germund
G. Dahlquist, A special stability problem for linear multistep
methods, Nordisk Tidskr. InformationsBehandling 3
(1963), 27–43. MR 0170477
(30 #715)
 [6]
Germund
Dahlquist, Error analysis for a class of methods for stiff
nonlinear initial value problems, Numerical analysis (Proc. 6th
Biennial Dundee Conf., Univ. Dundee, Dundee, 1975) Springer, Berlin,
1976, pp. 60–72. Lecture Notes in Math., Vol. 506. MR 0448898
(56 #7203)
 [7]
G. DAHLQUIST, On the Relation of GStability to Other Stability Concepts for Linear Multistep Methods, Report TRITANA7618, 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,
Neuchâtel, 1972), AkademieVerlag, Berlin, 1974,
pp. 125–131. Schr. Zentralinst. Math. Mech. Akad. Wiss. DDR,
Heft 20. MR
0364880 (51 #1134)
 [9]
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
(52 #9022)
 [10]
Olavi
Nevanlinna, On error bounds for 𝐺stable methods,
Nordisk Tidskr. Informationsbehandling (BIT) 16 (1976),
no. 1, 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 HTKKMATA76, Inst. of Math., Helsinki Univ. of Tech., 1976.
 [12]
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
0494953 (58 #13728)
 [13]
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 0399978
(53 #3816)
 [1]
 H. BREZIS, Opérateurs Maximaux Monotones, NorthHolland, Amsterdam, 1973. 2. H. BREZIS & A. PAZY, "Semi groups of nonlinear contractions on convex sets," J. Functional Analysis, v. 6, 1970, pp. 237281. MR 0448185 (56:6494)
 [3]
 R. BRUCK, "Asymptotic convergence of nonlinear contraction semi groups in Hilbert space," J. Functional Analysis, v. 18, 1975, pp. 1526. 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. 265298. MR 0287357 (44:4563)
 [5]
 G. DAHLQUIST, "A special stability problem for linear multistep methods," BIT, v. 3, 1963, pp. 2743. MR 0170477 (30:715)
 [6]
 G. DAHLQUIST, Error Analysis for a Class of Methods for Stiff NonLinear Initial Value Problems, Lecture Notes in Math., vol. 506, SpringerVerlag, Berlin and New York, 1976, pp. 6074. MR 0448898 (56:7203)
 [7]
 G. DAHLQUIST, On the Relation of GStability to Other Stability Concepts for Linear Multistep Methods, Report TRITANA7618, 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. Summerschool held in Oct. 1972 at Neuendorf), AkademieVerlag, Berlin, 1974, pp. 125131. 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. 147207. MR 0388185 (52:9022)
 [10]
 O. NEVANLINNA, "On error bounds for Gstable methods," BIT, v. 16, 1976, pp. 7984. MR 0488767 (58:8281)
 [11]
 O. NEVANLINNA, On Multistep Methods for Nonlinear Initial Value Problems with an Application to Minimization of Convex Functionals, Report HTKKMATA76, 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. 5871. 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. 96113. MR 0399978 (53:3816)
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
47H15,
65J05
Retrieve articles in all journals
with MSC:
47H15,
65J05
Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197805132039
PII:
S 00255718(1978)05132039
Article copyright:
© Copyright 1978
American Mathematical Society
