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 L^2 gradient flow for the Modica–Mortola regularization of the functional \[ v ∈ BV ( T d ; { − 1 , 1 } ) ↦ E ( v ) := γ 2 ∫ T d | ∇ v | + 1 2 ∑ k ∈ Z d σ ( k ) | v ^ ( k ) | 2 . v ∈ BV(T^d; \{-1,1\}) ↦ E(v) := γ /2 ∫ T^d |∇ v| + 1/2∑ k ∈ Z^d σ (k) |v̂(k)|^2. \] Here γ γ is the interfacial energy per unit length or unit area, T d T^d is the flat torus in R d