article
Shadowing for discrete approximations of abstract parabolic equations
published
yes
Wolf-Jürgen
Beyn
author 12477
Sergey
Piskarev
author
10020
department
110284488
department
This paper is devoted to the numerical analysis of abstract semilinear parabolic problems u'(t) - Au(t) + f(u(t)), u(0) - u(0), in some general Banach space E. We prove a shadowing Theorem that compares solutions of the continuous problem with those of a semidiscrete approximation (time stays continuous) in the neighborhood of a hyperbolic equilibrium. We allow rather general discretization schemes following the theory of discrete approximations developed by F. Stummel, R.D. Grigorieff and G. Vainikko. We use a compactness principle to show that the decomposition of the flow into growing and decaying solutions persists for this general type of approximation. The main assumptions of our results are naturally satisfied for operators with compact resolvents and can be verified for finite element as well as finite difference methods. In this way we obtain a unified approach to shadowing results derived e.g. in the finite element context ([ , , ]).
American Institute of Mathematical Sciences2008
eng
theory of shadowinghyperbolicequilibriasemidiscretizationcompact convergence of resolventsabstract parabolic problems
Discrete and Continuous Dynamical Systems-Series B
1531-3492
00025612170000210.3934/dcdsb.2008.10.19
10119-42
<div style="text-indent:-25px; padding-left:25px;padding-bottom:0px;">Beyn, W. - J., & Piskarev, S. (2008). Shadowing for discrete approximations of abstract parabolic equations. <em>Discrete and Continuous Dynamical Systems-Series B</em>, <em>10</em>(1), 19-42. doi:10.3934/dcdsb.2008.10.19</div>
Beyn W-J, Piskarev S. Shadowing for discrete approximations of abstract parabolic equations. <em>Discrete and Continuous Dynamical Systems-Series B</em>. 2008;10(1):19-42.
W.-J. Beyn and S. Piskarev, “Shadowing for discrete approximations of abstract parabolic equations”, <em>Discrete and Continuous Dynamical Systems-Series B</em>, vol. 10, 2008, pp. 19-42.
<div style="text-indent:-25px; padding-left:25px;padding-bottom:0px;">Beyn, Wolf-Jürgen, and Piskarev, Sergey. 2008. “Shadowing for discrete approximations of abstract parabolic equations”. <em>Discrete and Continuous Dynamical Systems-Series B</em> 10 (1): 19-42.</div>
Beyn W-J, Piskarev S (2008) <br />Shadowing for discrete approximations of abstract parabolic equations.<br />Discrete and Continuous Dynamical Systems-Series B 10(1): 19-42.
<div style="text-indent:-25px; padding-left:25px;padding-bottom:0px;">Beyn, W.-J. & Piskarev, S. (2008). Shadowing for discrete approximations of abstract parabolic equations. <em>Discrete and Continuous Dynamical Systems-Series B</em>, <em>10</em>(1), 19-42. American Institute of Mathematical Sciences. doi:10.3934/dcdsb.2008.10.19.</div>
Beyn, Wolf-Jürgen, and Piskarev, Sergey. “Shadowing for discrete approximations of abstract parabolic equations”. <em>Discrete and Continuous Dynamical Systems-Series B</em> 10.1 (2008): 19-42.
Beyn W-J, Piskarev S (2008) <br /><em>Discrete and Continuous Dynamical Systems-Series B</em> 10(1): 19-42.
Beyn, W. - J.; Piskarev, S. (2008): Shadowing for discrete approximations of abstract parabolic equations <em>Discrete and Continuous Dynamical Systems-Series B</em>,10:(1): 19-42.
Beyn, W.-J., & Piskarev, S., 2008. Shadowing for discrete approximations of abstract parabolic equations. <em>Discrete and Continuous Dynamical Systems-Series B</em>, 10(1), p 19-42.
Beyn, W. - J., & Piskarev, S. (2008). Shadowing for discrete approximations of abstract parabolic equations. <em>Discrete and Continuous Dynamical Systems-Series B</em>, <em>10</em>(1), 19-42. doi:10.3934/dcdsb.2008.10.19
W. - J. Beyn, and S. Piskarev, “Shadowing for discrete approximations of abstract parabolic equations”, <em>Discrete and Continuous Dynamical Systems-Series B</em>, <strong>2008</strong>, <em>10</em>, 19-42.
Beyn, W.-J., Piskarev, S.: Shadowing for discrete approximations of abstract parabolic equations. Discrete and Continuous Dynamical Systems-Series B. 10, 19-42 (2008).
Beyn, W. - J., and Piskarev, S. (2008). Shadowing for discrete approximations of abstract parabolic equations. <em>Discrete and Continuous Dynamical Systems-Series B</em> 10, 19-42.
15877402010-04-26T09:44:10Z2019-06-28T09:03:20Z