For equations of generalized Cahn–Hilliard type we present an a posteriori error analysis that is robust with respect to a small interface length scale γ. We propose the solution of a fourth order elliptic eigenvalue problem in each time step to gain a fully computable error bound, which only depends polynomially (of low order) on the inverse of γ. A posteriori and a priori error bounds for the eigenvalue problem are also derived. In numerical examples we demonstrate that this approach extends the applicability of robust a posteriori error estimation as it removes restrictive conditions on the initial data. Moreover we show that the computation of the principal eigenvalue allows the detection of critical points during the time evolution that limit the validity of the estimate.
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Sören Bartels, Rüdiger Müller, A posteriori error controlled local resolution of evolving interfaces for generalized Cahn–Hilliard equations. Interfaces Free Bound. 12 (2010), no. 1, pp. 45–74