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 L 2 \textrm {L}^2 gradient flow for the Modica–Mortola regularization of the functional \[ v ∈ BV ( T d ; { − 1 , 1 } ) ↦ E ( v ) := γ 2 ∫ T d | ∇ v | + 1 2 ∑ k ∈ Z d σ ( k ) | v ^ ( k ) | 2 . v \in \textrm {BV}(\mathbb {T}^d; \{-1,1\}) \mapsto E(v) := \frac {\gamma }{2} \int _{\mathbb {T}^d} |abla v| + \frac {1}{2}\sum _{k \in \mathbb {Z}^d} \sigma (k) |\hat {v}(k)|^2. \] Here γ \gamma is the interfacial energy per unit length or unit area, T d \mathbb {T}^d is the flat torus in R d \mathbb {R}^d , and σ \sigma is a nonnegative Fourier multiplier, that is continuous on R d \mathbb {R}^d , symmetric in the sense that σ ( ξ ) = σ ( − ξ ) \sigma (\xi )=\sigma (-\xi ) for all ξ ∈ R d \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 ℓ ∞ ( 0 , T ; L 2 ( T d ) ) \ell ^\infty (0,T;\mathrm {L}^2(\mathbb {T}^d)) norm.