Robust a posteriori error analysis of the stochastic Cahn-Hilliard equation with rough noise

We derive a posteriori error estimate for a fully discrete adaptive finite element approximation of the stochastic Cahn-Hilliard equation with rough noise. The considered model is derived from the stochastic Cahn-Hilliard equation with additive space-time white noise through suitable spatial regularization of the white noise. The a posteriori estimate is robust with respect to the interfacial width parameter as well as the noise regularization parameter. We propose a practical adaptive algorithm for the considered problem and perform numerical simulations to illustrate the theoretical findings.

💡 Research Summary

The paper addresses a major gap in the numerical analysis of the stochastic Cahn‑Hilliard equation driven by additive space‑time white noise. Because the white noise is too irregular, the solution lacks the H² spatial regularity required for standard a‑posteriori error analysis. The authors overcome this obstacle by first regularizing the noise: they replace the white noise with a piecewise‑linear finite‑element projection (equation (4)), which preserves the zero‑mean property and yields a noise that is only H¹‑regular.

With the regularized problem (5) in hand, the solution u is split into a linear stochastic component ũ solving a linear SPDE (6) and a deterministic‑random component b̂ solving a random PDE (7). For the linear part, a modified linear transformation y = ũ − ∫₀ᵗ df_W(s) (equation (9)) is introduced. Unlike previous works that required H⁴‑regular noise, this transformation works with H¹‑regular noise and leads to a deterministic‑type PDE (11) for y, ensuring that ∂ₜy∈L²(0,T;H⁻¹) almost surely.

The nonlinear random PDE is treated path‑wise. The authors define three high‑probability events: Ω_ε (control of the linear SPDE error), Ω_{δ,ε} (bounds on L^∞(0,T;H⁻¹) and L⁴(0,T;L⁴) norms), and Ω_{γ,ε} (bound on L^∞(0,T;L⁴) norm). Lemmas A.3 and A.4, together with a new interpolation inequality (Lemma A.1), show that each event has probability at least 1 − C ε for any prescribed ε>0. Consequently, the path‑wise a‑posteriori estimate (Theorem 6.1) holds on the intersection of these events, i.e. with arbitrarily high probability.

On the discretization side, a fully discrete scheme is proposed. Spatially, piecewise‑linear continuous finite elements V_{n}^{h} are used on possibly non‑uniform meshes; temporally, arbitrary time steps τ_n are allowed. The scheme (15) couples the discrete order‑parameter u_{n}^{h} and chemical potential w_{n}^{h}. The linear part (17) and the nonlinear part (18) are solved separately, mirroring the continuous splitting.

Residual‑based error indicators are defined for both space and time: η_{SPACE,1}, η_{SPACE,2}, η_{SPACE,3} capture element‑wise time‑derivative residuals, jumps of ∇w·n, and jumps of ∇u·n respectively; η_{TIME,1}, η_{TIME,2}, η_{TIME,3} measure differences between successive time levels. Lemma 5.2 shows that the residuals R_y and S_y can be bounded by linear combinations of these indicators, with constants depending only on mesh regularity.

The error analysis proceeds in two stages. First, for the linear SPDE, Lemma 5.6 and Corollary B.1 provide explicit convergence rates O(h)+O(τ^{1/2}) even with H¹‑regular noise. Second, for the nonlinear RPDE, Theorem 6.1 gives a path‑wise bound in terms of the residual indicators, valid on the high‑probability set. By combining these results, Theorem 7.1 delivers a global a‑posteriori error estimate for the fully discrete approximation of the regularized stochastic Cahn‑Hilliard equation. Importantly, the bound is robust: the constants do not blow up as the interfacial width ε→0 nor as the noise regularization parameter varies. This improves upon earlier work