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Title: A-stable time discretizations preserve maximal parabolic regularity
Authors: Kovács, B
Li, B
Lubich, C
Keywords: A-stability
Maximal regularity
Multistep methods
Parabolic equations
Runge-Kutta methods
Issue Date: 2016
Publisher: Society for Industrial and Applied Mathematics
Source: SIAM journal on numerical analysis, 2016, v. 54, no. 6, p. 3600-3624 How to cite?
Journal: SIAM journal on numerical analysis 
Abstract: It is shown that for a parabolic problem with maximal Lp-regularity (for 1 < p < ∞), the time discretization by a linear multistep method or Runge-Kutta method has maximal ℓp-regularity uniformly in the stepsize if the method is A-stable (and satisfies minor additional conditions). In particular, the implicit Euler method, the Crank-Nicolson method, the second-order backward difference formula (BDF), and the Radau IIA and Gauss Runge-Kutta methods of all orders preserve maximal regularity. The proof uses Weis' characterization of maximal Lp-regularity in terms of R-boundedness of the resolvent, a discrete operator-valued Fourier multiplier theorem by Blunck, and generating function techniques that have been familiar in the stability analysis of time discretization methods since the work of Dahlquist. The A(α)-stable higher-order BDF methods have maximal ℓp-regularity under an R-boundedness condition in a larger sector. As an illustration of the use of maximal regularity in the error analysis of discretized nonlinear parabolic equations, it is shown how error bounds are obtained without using any growth condition on the nonlinearity or for nonlinearities having singularities.
ISSN: 0036-1429
EISSN: 1095-7170
DOI: 10.1137/15M1040918
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