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DOI: 10.1137/090757940

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SIAM Journal on Numerical Analysis 48, 6 (2010) 2019-2042

Explicit Runge--Kutta schemes and finite elements with symmetric stabilization for first-order linear PDE systems

Erik Burman 1, Alexandre Ern 2, Miguel Ángel Fernández 3

(2010)

We analyze explicit Runge--Kutta schemes in time combined with stabilized finite elements in space to approximate evolution problems with a first-order linear differential operator in space of Friedrichs-type. For the time discretization, we consider explicit second- and third-order Runge--Kutta schemes. We identify a general set of properties on the spatial stabilization, encompassing continuous and discontinuous finite elements, under which we prove stability estimates using energy arguments. Then, we establish $L^2$-norm error estimates with (quasi-)optimal convergence rates for smooth solutions in space and time. These results hold under the usual CFL condition for third-order Runge--Kutta schemes and any polynomial degree in space and for second-order Runge--Kutta schemes and first-order polynomials in space. For second-order Runge--Kutta schemes and higher polynomial degrees in space, a tightened 4/3-CFL condition is required. Numerical results are presented for the advection and wave equations.

1:	Department of Mathematics
	University of Sussex
2:	Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS)
	Ecole des Ponts ParisTech
3:	REO (INRIA Rocquencourt)
	INRIA – Laboratoire Jacques-Louis Lions

Subject

:

Mathematics/Numerical Analysis

Keyword(s): First-order PDEs – transient problems – stabilized finite elements – explicit Runge--Kutta schemes – stability – convergence

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From: Alexandre Ern <>

Submitted on: Monday, 4 May 2009 12:21:45

Updated on: Friday, 4 November 2011 12:01:45