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IMA Journal of Numerical Analysis 2003 23(3):465-487; doi:10.1093/imanum/23.3.465
© 2003 by Institute of Mathematics and its Applications
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Smoothing properties and approximation of time derivatives for parabolic equations: constant time steps

Yubin Yan1

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

We study smoothing properties and approximation of time derivatives for time discretization schemes with constant time steps for a homogeneous parabolic problem formulated as an abstract initial-value problem in a Banach space. The time stepping schemes are based on using rational functions r(z) {approx} e–z which are A({theta})-stable for suitable {theta} [0, {pi}/2] and satisfy |r({infty})| < 1, and the approximations of time derivatives are based on using difference quotients in time. Both smooth and non-smooth data error estimates of optimal order for the approximation of time derivatives are proved. Further, we apply the results to obtain error estimates of time derivatives in the supremum norm for fully discrete methods based on discretizing the spatial variable by a finite-element method.

Key Words: Banach space; parabolic; smoothing; time derivative; single step time stepping methods; fully discrete schemes; error estimates; finite-element methods

Received 26 February 2002. Revised 15 November 2002.


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