Spectral approximation of pattern-forming nonlinear evolution equations with double-well potentials of quadratic growth
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- by Nicolas Condette, Christof Melcher and Endre Süli;
- Math. Comp. 80 (2011), 205-223
- DOI: https://doi.org/10.1090/S0025-5718-10-02365-3
- Published electronically: May 5, 2010
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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 \[ v \in \textrm {BV}(\mathbb {T}^d; \{-1,1\}) \mapsto E(v) := \frac {\gamma }{2} \int _{\mathbb {T}^d} |\nabla v| + \frac {1}{2}\sum _{k \in \mathbb {Z}^d} \sigma (k) |\hat {v}(k)|^2.\] 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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Bibliographic 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
- Received by editor(s): April 23, 2009
- Received by editor(s) in revised form: September 23, 2009
- Published electronically: May 5, 2010
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 80 (2011), 205-223
- MSC (2010): Primary 65M70; Secondary 82D40, 82D60
- DOI: https://doi.org/10.1090/S0025-5718-10-02365-3
- MathSciNet review: 2728977