*Maximal Parabolic Regularity Under Time Discretization*,

**Christian Lubich**

Στο πλαίσιο της διοργάνωσης των σεμιναρίων του τμήματος, θα πραγματοποιηθεί την Παρασκευή 30/10/2015 και ώρα 12:00 στην αίθουσα Σεμιναρίων του Τμήματος Μηχανικών Η/Υ και Πληροφορικής, ομιλία με τίτλο "Maximal Parabolic Regularity Under Time Discretization". Ομιλητής θα είναι ο κ. Christian Lubich, Professor, University of Tübingen, Germany.

ΠΕΡΙΛΗΨΗMaximal regularity is an important mathematical tool in studying existence, uniqueness and regularity of the solution of nonlinear parabolic partial differential equations. In this talk the following question is addressed:

Given an operator on a Banach space that has maximal Lp-regularity (for 1 < p < ∞), for which (if any) time discretization methods for the associated parabolic initial value problem is the maximal regularity preserved in the discrete ℓp-setting, uniformly in the stepsize?

It is found that 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). The notion of A-stability was introduced by Dahlquist in 1963 and is a well-known concept in the study of time discretization methods. 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.

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