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

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.