Optimal convergence rate for the nonlinear Cahn–Hilliard–Cook equation

Obtain an optimal rate of strong convergence for the spatially semidiscrete finite element approximation of the nonlinear Cahn–Hilliard–Cook equation.

Background

The paper analyzes finite element and implicit Euler approximations for the linearized Cahn–Hilliard–Cook equation and interprets the results as approximation results for the stochastic convolution appearing in the mild solution of the nonlinear equation. It notes that strong convergence for the spatially semidiscrete approximation of the nonlinear Cahn–Hilliard–Cook equation has been proved elsewhere, but without a known convergence rate. The authors identify establishing the optimal rate as an unresolved challenge.

References

The remaining part, which solves a nonlinear random evolution problem, is studied in \citet{KLMchc}, where strong convergence is proved for the spatially semidiscrete approximation of the nonlinear Cahn-Hilliard-Cook equation, but without known rate of convergence. To obtain the optimal rate of convergence remains a challenge.

— Finite element approximation of the linearized Cahn-Hilliard-Cook equation  (2608.12833 - Mesforush et al., 13 Aug 2026) in Section 4, Conclusions

To prove strong convergence with an estimate of the rate remains a challenge for future work.

— Finite Element Approximation of the Cahn-Hilliard-Cook equation  (2608.16680 - Mesforush et al., 17 Aug 2026) in Section 1, Introduction