Finite element error estimates for 3D exterior incompressible flow with nonzero velocity at infinity

Abstract  We consider a stationary incompressible Navier–Stokes flow in a 3D exterior domain, with nonzero velocity at infinity. In
order to approximate this flow, we use the stabilized P1–P1 finite element method proposed by Rebollo (Numer Math 79:283–319,
1998). Following an approach by Guirguis and Gunzburger (Model Math Anal Numer 21:445–464, 1987), we apply this method to
the Navier–Stokes system with Oseen term in a truncated exterior domain, under a pointwise boundary condition on the artificial
boundary. This leads to a discrete problem whose solution approximates the exterior flow, as is shown by error estimates.

• Content Type Journal Article
• DOI 10.1007/s00211-009-0253-3
• Authors
  
  
  • Paul Deuring, Univ Lille Nord de France 59000 Lille France
  
  

• Journal Numerische Mathematik
• Online ISSN 0945-3245
• Print ISSN 0029-599X

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Original source : http://www.springerlink.com/content/913852x8538151...

Birkhoff normal form for splitting methods applied to semilinear Hamiltonian PDEs. Part I. Finite-dimensional discretization

Abstract  We consider discretized Hamiltonian PDEs associated with a Hamiltonian function that can be split into a linear unbounded operator and a regular
nonlinear part. We consider splitting methods associated with this decomposition. Using a finite-dimensional Birkhoff normal
form result, we show the almost preservation of the actions of the numerical solution associated with the splitting method over arbitrary long time and for asymptotically large level
of space approximation, provided the Sobolev norm of the initial data is small enough. This result holds under generic non-resonance conditions on the frequencies of the linear operator and on the step size. We apply these results to nonlinear
Schrödinger equations as well as the nonlinear wave equation.

• Content Type Journal Article
• DOI 10.1007/s00211-009-0258-y
• Authors
  
  
  • Erwan Faou, INRIA and Ecole Normale Supérieure de Cachan, Bretagne Avenue Robert Schumann 35170 Bruz France
  • Benoît Grébert, Université de Nantes Laboratoire de Mathématiques Jean Leray 2 rue de la Houssinière 44322 Nantes France
  • Eric Paturel, Université de Nantes Laboratoire de Mathématiques Jean Leray 2 rue de la Houssinière 44322 Nantes France
  
  

• Journal Numerische Mathematik
• Online ISSN 0945-3245
• Print ISSN 0029-599X

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Original source : http://www.springerlink.com/content/k11508g52p710w...

A finite element method for surface PDEs: matrix properties

Abstract  We consider a recently introduced new finite element approach for the discretization of elliptic partial differential equations
on surfaces. The main idea of this method is to use finite element spaces that are induced by triangulations of an “outer”
domain to discretize the partial differential equation on the surface. The method is particularly suitable for problems in
which there is a coupling with a problem in an outer domain that contains the surface, for example, two-phase flow problems.
It has been proved that the method has optimal order of convergence both in the H
¹ and in the L
²-norm. In this paper, we address linear algebra aspects of this new finite element method. In particular the conditioning
of the mass and stiffness matrix is investigated. For the two-dimensional case we present an analysis which proves that the
(effective) spectral condition number of the diagonally scaled mass matrix and the diagonally scaled stiffness matrix behaves
like h
⁻³| ln h| and h
⁻²| ln h|, respectively, where h is the mesh size of the outer triangulation.

• Content Type Journal Article
• DOI 10.1007/s00211-009-0260-4
• Authors
  
  
  • Maxim A. Olshanskii, Moscow State M.V. Lomonosov University Department of Mechanics and Mathematics 119899 Moscow Russia
  • Arnold Reusken, RWTH-Aachen University Institut für Geometrie und Praktische Mathematik 52056 Aachen Germany
  
  

• Journal Numerische Mathematik
• Online ISSN 0945-3245
• Print ISSN 0029-599X

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Original source : http://www.springerlink.com/content/a4n187ur25564k...

Continuous two-step Runge–Kutta methods for ordinary differential equations

Abstract  New classes of continuous two-step Runge-Kutta methods for the numerical solution of ordinary differential equations are derived.
These methods are developed imposing some interpolation and collocation conditions, in order to obtain desirable stability
properties such as A-stability and L-stability. Particular structures of the stability polynomial are also investigated.

• Content Type Journal Article
• Category Original Paper
• DOI 10.1007/s11075-009-9329-5
• Authors
  
  
  • Raffaele D’Ambrosio, Universitá degli Studi di Salerno Dipartimento di Matematica e Informatica Fisciano (SA) 84084 Italy
  • Zdzislaw Jackiewicz, Arizona State University Department of Mathematics and Statistics Tempe AZ 85287 USA
  
  

• Journal Numerical Algorithms
• Online ISSN 1572-9265
• Print ISSN 1017-1398

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Original source : http://www.springerlink.com/content/f6228520840626...