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 L 2 gradient flow for the Modica-Mortola regularization of the functional v ∈ BV(T d ; {-1,1}) ↦ E(v) := γ/2 ∫ Td ∇v + 1/2 ∑ k∈ℤd σ(k)v(k) 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 ∈ 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 )) norm.