On a posteriori error control for the Allen–Cahn problem

摘要

We derive a posteriori bounds in non-Hilbertian norms for finite element approximations to the Allen–Cahn phase field model. The derivation of the bounds relies crucially on an approximate spectral estimate for the linearised steady part of the partial differential equation, yielding only polynomial dependence on the phase separation length. A second key ingredient in the proof of the a posteriori bounds is the use of the elliptic reconstruction framework, enabling the control of the error in the L~∞1(L~r)-norms, r∈[2,∞]. For simplicity of the presentation, only the spatially discrete method is considered here.