Spectral approximation of pattern-forming nonlinear evolution equations with double-well potentials of quadratic growth. - Condette, N; Melcher, C; Süli, E (2011)
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 Here γ is the interfacial energy per unit length or unit area, Td is the flat torus in Td, and σ is a nonnegative Fourier multiplier, that is continuous on ℝd, symmetric in the sense that σ(ξ) = σ(-ξ) for all ξ ∈ℝd 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 (Td)) norm. © 2010 American Mathematical Society.