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

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

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

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.