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Smoothing Properties and Approximation of Time Derivatives for Parabolic Equations: Variable Time Steps

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Abstract

We study smoothing properties and approximation of time derivatives for time discretization schemes with variable time steps for a homogeneous parabolic problem formulated as an abstract initial value problem in a Banach space. The time stepping methods are based on using rational approximations to the exponential function which are A(Θ)-stable for suitable Θ∈(0,π/2] with unit bounded maximum norm. First- and second-order approximations of time derivatives based on using difference quotients are considered. Smoothing properties are derived and error estimates are established under the so-called increasing quasi-quasiuniform assumption on the time steps.

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REFERENCES

  1. N. Yu. Bakaev, On the bounds of approximations of holomorphic semigroups, BIT, 35 (1995), pp. 605-608.

    Google Scholar 

  2. G. A. Baker, J. H. Bramble,and V. Thomée, Single step Galerkin approximations for parabolic problems, M ath. Comp., 31 (1977), pp. 818-847.

    Google Scholar 

  3. M. Crouzeix, S. Larsson, S. Piskarëv,and V. Thomée, The stability of rational approximations of analytic semigroups, BIT, 33 (1993), pp. 74-84.

    Google Scholar 

  4. K. Eriksson, C. Johnson, and S. Larsson, Adaptive finite element methods for parabolic problems. VI. Analytic semigroups, SIA M J. Numer. Anal., 35 (1998), pp. 1315-1325.

    Google Scholar 

  5. H. Fujita and T. Suzuki, Evolution problems, in Handbook of Numerical Analysis, Vol. II, North-Holland, Amsterdam, 1991, pp. 789-928.

    Google Scholar 

  6. A. Hansbo, Nonsmooth data error estimates for damped single step methods for parabolic equations in Banach space, Calcolo, 36 (1999), pp. 75-101.

    Google Scholar 

  7. S. Larsson, V. Thomée, and L. B. Wahlbin, Finite-element methods for a strongly damped wave equation, IM A J. Numer. Anal., 11 (1991), pp. 115-142.

    Google Scholar 

  8. M.-N. Le Roux, Semidiscretization in time for parabolic problems, Math. Comp., 33 (1979), pp. 919-931.

    Google Scholar 

  9. C. Palencia, A stability result for sectorial operators in Banach spaces, SIAM J. Numer. Anal., 30 (1993), pp. 1373-1384.

    Google Scholar 

  10. C. Palencia, On the stability of variable stepsize rational approximations of holomorphic semigroups, Math. Comp., 62 (1994), pp. 93-103.

    Google Scholar 

  11. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.

    Google Scholar 

  12. V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer-Verlag, Berlin, 1997.

    Google Scholar 

  13. Y. Yan, Approximation of time derivatives for parabolic equations in Banach space: Constant time steps, IMA J. Numer. Anal., 23 (2003), pp. 465-487.

    Google Scholar 

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  1. Department of Mathematics, Chalmers University of Technology and Göteborg University, SE-412 96, Göteborg, Sweden

    Yubin Yan

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  1. Yubin Yan
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Yan, Y. Smoothing Properties and Approximation of Time Derivatives for Parabolic Equations: Variable Time Steps. BIT Numerical Mathematics 43, 647–669 (2003). https://doi.org/10.1023/B:BITN.0000007061.40742.4e

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