Displaying similar documents to “Irregularities in the numerical treatment of nonlinear initial value problems”

Remarks on Gårding inequalities for differential operators

Xavier Saint Raymond (2002)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Classical Gårding inequalities such as those of Hörmander, Hörmander-Melin or Fefferman-Phong are proved by pseudo-differential methods which do not allow to keep a good control on the supports of the functions under study nor on the smoothness of the coefficients of the operator. In this paper, we show by very simple calculations that in certain special situations, the results that can be obtained directly are much better than those expected thanks to the general theory.

Averages of uniformly continuous retractions

A. Jiménez-Vargas, J. Mena-Jurado, R. Nahum, J. Navarro-Pascual (1999)

Studia Mathematica

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Let X be an infinite-dimensional complex normed space, and let B and S be its closed unit ball and unit sphere, respectively. We prove that the identity map on B can be expressed as an average of three uniformly retractions of B onto S. Moreover, for every 0≤ r < 1, the three retractions are Lipschitz on rB. We also show that a stronger version where the retractions are required to be Lipschitz does not hold.

Layering methods for nonlinear partial differential equations of first order

Avron Douglis (1972)

Annales de l'institut Fourier

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This paper is concerned with generalized, discontinuous solutions of initial value problems for nonlinear first order partial differential equations. “Layering” is a method of approximating an arbitrary generalized solution by dividing its domain, say a half-space t ≥ 0 , into thin layers ( i - ℓ ) h ≤ t ≤ i h , i = 1 , 2 , ... ( h &gt; 0 ) , and using a strict solution u i in the i -th layer. On the interface t = ( i - ℓ ) h , u t is required to reduce to a smooth function approximating the values on that plane of u i - ℓ . The resulting stratified configuration...

Convergence of the finite element method applied to an anisotropic phase-field model

Erik Burman, Daniel Kessler, Jacques Rappaz (2004)

Annales mathématiques Blaise Pascal

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We formulate a finite element method for the computation of solutions to an anisotropic phase-field model for a binary alloy. Convergence is proved in the H 1 -norm. The convergence result holds for anisotropy below a certain threshold value. We present some numerical experiments verifying the theoretical results. For anisotropy below the threshold value we observe optimal order convergence, whereas in the case where the anisotropy is strong the numerical solution to the phase-field equation...