zbMATH — the first resource for mathematics

Error estimates over infinite intervals of some discretizations of evolution equations. (English) Zbl 0573.65038
The author considers the initial value problem in a reflexive B-space \(V: du/dt+F(t,u)=0\), \(u(0)=u_ 0\in V\), \(t>0\). The operator \(F: V\to V'\) is not necessarily bounded. Four cases are investigated: F monotone, F strongly monotone, F conservative, F non-monotone (the case (F(t,u)- F(t,v), u-v)\(\leq -\rho_ 0\| u-v\|^ 2\), \(\forall u,v\in V\) with \(\rho_ 0>0)\). The (implicit) \(\vartheta\)-method is applied: \(v(t-k)- v(t)+kF(t+(1-\vartheta)k,\vartheta v(t)+(1-\vartheta)v(t+k))=0\) with \(0\leq \vartheta \leq 1\). It is assumed that this nonlinear equation has a unique solution in V.
In each of the discussed cases estimates for the error \(e(t)=u(t)-v(t)\) are given in terms of e(0) and of the truncation error \[ R_{\vartheta}(t,u)=F(t+(1-\vartheta)k,\vartheta u(t)+(1- \vartheta)u(t+k))-k^{-1}\int^{t+k}_{t}F(s,u(s))ds. \] Many remarks on the behaviour of solutions are joint, also the so called ’forced solution’ (slowly increasing) in the non-monotone case is discussed. One of the sections of the paper is devoted to estimates of the truncation error. The last section discusses the so called boundary value techniques, in the context of the forced solution for the non-monotone case. The method presented here is composed of the repeated midpoint formula and the backward Euler on the last step only.
Reviewer: K.Moszyński

65J15 Numerical solutions to equations with nonlinear operators (do not use 65Hxx)
65L05 Numerical methods for initial value problems
34G20 Nonlinear differential equations in abstract spaces
Full Text: DOI
[1] O. Axelsson,Error estimates for Galerkin methods for quasilinear parabolic and elliptic differential equations in divergence form, Numer. Math. 28, 1–14 (1977). · Zbl 0341.65067 · doi:10.1007/BF01403853
[2] O. Axelsson and T. Steihaug,Some computational aspects in the numerical solution of parabolic equations, J. Comp. Appl. Math. 4 (1978), 129–142. · Zbl 0409.65043 · doi:10.1016/0771-050X(78)90037-2
[3] O. Axelsson,Global integration of differential equations through Lobatto quadrature, BIT 4 (1964), 69–86. · Zbl 0122.12204 · doi:10.1007/BF01939850
[4] O. Axelsson and J. G. Verwer,Boundary value techniques for initial value problems in ordinary differential equations, Technical report (1983), Mathematical Centre, Amsterdam, The Netherlands. · Zbl 0586.65052
[5] G. Dahlquist,Stability and error bounds in the numerical integration of ordinary differential equations (Thesis), Transactions of the Royal Institute of Technology, No. 130, Stockholm, 1959. · Zbl 0085.33401
[6] G. Dahlquist,Error analysis for a class of methods for stiff non-linear initial value problems, inNumerical Analysis (G. A. Watson, ed.), Dundee 1975, Springer-Verlag, LN M506, 1976.
[7] R. Frank, J. Schneid and C. W. Ueberhuber,The concept of B-convergence, SIAM J. Numer. Anal. 18 (1981), 753–780. · Zbl 0467.65032 · doi:10.1137/0718051
[8] A. Friedman,Partial Differential Equations, Holt, Rinehart and Winston, Inc. New York, 1969. · Zbl 0224.35002
[9] P. Henrici,Discrete Variable Methods in Ordinary Differential Equations, John Wiley and Sons Inc., New York, 1962. · Zbl 0112.34901
[10] O. Karlquist,Numerical solution of elliptic differential equations, Tellus (1952), 374–384.
[11] J. D. Lambert,Computational Methods in Ordinary Differential Equations. Wiley, London, 1973. · Zbl 0258.65069
[12] W. J. Layton and R. M. M. Mattheij,Estimates over infinite intervals of approximations to initial value problems, Technical report, Department of Mathematics, Catholic University, Nijmegen, 1983.
[13] S. Schechter,Quasi-tridiagonal matrices and type-insensitive difference equations, Quart. Appl. Math. 18 (1960), 285–295. · Zbl 0097.32902
[14] J. G. Verwer and K. Dekker,Step-by-step stability in the numerical solution of partial differential equations, Technical report, 1983, Center for Math. and Comp. Sc., Kruislaan 413, Amsterdam. · Zbl 0512.65054
[15] G. Wanner,A short proof on nonlinear A-stability, BIT 16 (1976), 226–227. · Zbl 0329.65048 · doi:10.1007/BF01931374
[16] O. C. Zienkiewicz,The Finite Element Method in Engineering, Science, McGraw Hill, London, 1971. · Zbl 0237.73071
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.