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

Journal article published in 2010 by Nicolas Condette, Christof Melcher, Endre Süli ORCID
This paper is available in a repository.
This paper is available in a repository.

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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 L² gradient flow for the Modica-Mortola regularization of the functional $υ \, 𝟄 \,BV(𝕋^d ;\,\left\{ { - 1,1} \right\})\, ↦ \,E(υ )\,: = \,\frac{\gamma } {2}∫_{𝕋^d } {\,\left| {∇ υ } \right|\, + \,\frac{1} {2}} ∑\limits_{k 𝟄 ℤ^d } {σ (k)\left| {\hat υ \left( k \right)} \right|^2 } $ Here γ is the interfacial energy per unit length or unit area, 𝕋 d is the flat torus in ℝ d , and σ is a nonnegative Fourier multiplier, that is continuous on ℝ d , symmetric in the sense that σ(ξ) = σ (—ξ) for all ξ Є 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 l ∞ (0, T; L² (𝕋 d )) norm.