A-stable time discretizations preserve maximal parabolic regularity

Balázs Kovács, Buyang Li, Christian Lubich

Research output: Journal article publicationJournal articleAcademic researchpeer-review

29 Citations (Scopus)

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.
Original languageEnglish
Pages (from-to)3600-3624
Number of pages25
JournalSIAM Journal on Numerical Analysis
Volume54
Issue number6
DOIs
Publication statusPublished - 1 Jan 2016

Keywords

  • A-stability
  • Maximal regularity
  • Multistep methods
  • Parabolic equations
  • Runge-Kutta methods

ASJC Scopus subject areas

  • Numerical Analysis

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