Equadiff 6 pp 275-284 | Cite as

Stability and error estimates valid for infinite time, for strongly monotone and infinitely stiff evolution equations

  • O. Axelsson
Lectures Presented In Sections Section C Numerical Methods
Part of the Lecture Notes in Mathematics book series (LNM, volume 1192)


For evolution equation with a monotone operator we derive unconditional stability and error estimates valid for all times. θ-method, For the θ=1/2(2+ζτV), 0<v≤1, ζ>0), we prove an error estimate O(τ4/3), τ → 0, if V=1/3, where τ is the maximal timestep for an arbitrary choice of the sequence of timesteps and with no further condition on F, and an estimate O(τ2) under some additional conditions. The first result is an improvement over the implicit midpoint method (θ=½), for which an order reduction to O(τ) may occur.


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  1. 1.
    O. Axelsson, Error estimates for Galerkin methods for quasilinear parabolic and elliptic differential equations in divergence form, Numer. Math. 28, 1–14(1977).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    O. Axelsson, Error estimates over infinite intervals of some discretizations of evolution equations, BIT 24(1984), 413–424.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    G. Dahlquist, Error analysis for a class of methods for stiff nonlinear initial value problems, Numerical Analysis (G.A. Watson, ed.), Dundee 1975, Springer-Verlag, LNM 506, 1976.Google Scholar
  4. 4.
    R. Frank, J. Schneid and C.W. Ueberhuber, The concept of B-convergence, SIAM J. Numer. Anal. 18(1981), 753–780.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    J. Kraaijevanger, B-convergence of the implicit midpoint rule and the trapezoidal rule, Report no. 01-1985, Institute of Applied Mathematics and Computer Science, University of Leiden, The Netherlands.Google Scholar
  6. 6.
    A. Prothero and A. Robinson, The stability and accuracy of one-step methods, Math. Comp. 28(1974), 145–162.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    K. Rektorys, The Method of Discretization in Time and Partial Differential Equations. D. Reidel Publ. Co., Dordrecht-Holland, Boston-U.S.A., 1982.zbMATHGoogle Scholar

Copyright information

© Equadiff 6 and Springer-Verlag 1986

Authors and Affiliations

  • O. Axelsson
    • 1
  1. 1.Department of MathematicsUniversity of Nijmegen ToernooiveldNijmegenThe Netherlands

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