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Spectral approximation of pattern-forming nonlinear evolution equations with double-well potentials of quadratic growth


Authors: Nicolas Condette, Christof Melcher and Endre Süli
Journal: Math. Comp. 80 (2011), 205-223
MSC (2010): Primary 65M70; Secondary 82D40, 82D60
Published electronically: May 5, 2010
MathSciNet review: 2728977
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Abstract: This paper is concerned with the analysis of a numerical algorithm for the approximate solution of a class of nonlinear evolution problems that arise as $ \textrm{L}^2$ gradient flow for the Modica-Mortola regularization of the functional

$\displaystyle v \in \textrm{BV}(\mathbb{T}^d; \{-1,1\}) \mapsto E(v) := \frac{... ...\vert + \frac{1}{2}\sum_{k \in \mathbb{Z}^d} \sigma(k) \vert\hat{v}(k)\vert^2.$

Here $ \gamma$ is the interfacial energy per unit length or unit area, $ \mathbb{T}^d$ is the flat torus in $ \mathbb{R}^d$, and $ \sigma$ is a nonnegative Fourier multiplier, that is continuous on $ \mathbb{R}^d$, symmetric in the sense that $ \sigma(\xi)=\sigma(-\xi)$ for all $ \xi \in \mathbb{R}^d$ and that decays to zero at infinity.

Such functionals feature in mathematical models of pattern-formation in micromagnetics and models of diblock copolymers. The resulting evolution equation is discretized by a Fourier spectral method with respect to the spatial variables and a modified Crank-Nicolson scheme in time. Optimal-order a priori bounds are derived on the global error in the $ \ell^\infty(0,T;\mathrm{L}^2(\mathbb{T}^d))$ norm.


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Additional Information

Nicolas Condette
Affiliation: Department of Mathematics, Humboldt-Universität zu Berlin, Unter den Linden 6, D-10099 Berlin, Germany
Email: condette@mathematik.hu-berlin.de

Christof Melcher
Affiliation: Department of Mathematics I, RWTH Aachen University, D-52056 Aachen, Germany
Email: melcher@math1.rwth-aachen.de

Endre Süli
Affiliation: Mathematical Institute, University of Oxford, 24-29 St Giles’, Oxford OX1 3LB, United Kingdom.
Email: endre.suli@maths.ox.ac.uk

DOI: https://doi.org/10.1090/S0025-5718-10-02365-3
Received by editor(s): April 23, 2009
Received by editor(s) in revised form: September 23, 2009
Published electronically: May 5, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.