Spectral approximation of pattern-forming nonlinear evolution equations with double-well potentials of quadratic growth
Nicolas Condette, Christof Melcher, ENDRE E. SÜLI · Mathematics of Computation · 2010
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