Advances in Differential Equations

Convergence of a semidiscrete scheme for a forward-backward parabolic equation

Giovanni Bellettini, Carina Geldhauser, and Matteo Novaga

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Abstract

We study the convergence of a semidiscrete scheme for the forward-backward parabolic equation $u_t= (W'(u_x))_x$ with periodic boundary conditions in one space dimension, where $W$ is a standard double-well potential. We characterize the equation satisfied by the limit of the discretized solutions as the grid size goes to zero. Using an approximation argument, we show that it is possible to flow initial data ${\overline u}$ having regions where ${\overline u}_x$ falls within the concave region $\{W''<0\}$ of $W$, where the backward character of the equation manifests itself. It turns out that the limit equation depends on the way we approximate ${\overline u}$ in its unstable region. Our result can be viewed as a characterization, among all Young measure solutions of the equation, of the much smaller subset of those solutions which can be obtained as limit of the semidiscrete scheme.

Article information

Source
Adv. Differential Equations Volume 18, Number 5/6 (2013), 495-522.

Dates
First available in Project Euclid: 14 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.ade/1363266255

Mathematical Reviews number (MathSciNet)
MR3086463

Zentralblatt MATH identifier
1307.65118

Subjects
Primary: 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 35K20: Initial-boundary value problems for second-order parabolic equations 35K55: Nonlinear parabolic equations 65M12: Stability and convergence of numerical methods

Citation

Bellettini, Giovanni; Geldhauser, Carina; Novaga, Matteo. Convergence of a semidiscrete scheme for a forward-backward parabolic equation. Adv. Differential Equations 18 (2013), no. 5/6, 495--522. https://projecteuclid.org/euclid.ade/1363266255.


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