Error estimates for fully discrete BDF finite element approximations of the Allen–Cahn equation
Georgios Akrivis, Buyang Li
Abstract
Abstract For a class of compatible profiles of initial data describing bulk phase regions separated by transition zones, we approximate the Cauchy problem of the Allen–Cahn (AC) phase field equation by an initial-boundary value problem in a bounded domain with the Dirichlet boundary condition. The initial-boundary value problem is discretized in time by the backward difference formulae (BDF) of order $1\leqslant q\leqslant 5$ and in space by the Galerkin finite element method of polynomial degree $r-1$, with $r\geqslant 2$. We establish an error estimate of $O(\tau ^q\varepsilon ^{-q-\frac 12}+h^{r}\varepsilon ^{-r-\frac 12}+{e}^{-c/\varepsilon })$ with explicit dependence on the small parameter $\varepsilon$ describing the thickness of the phase transition layer. The analysis utilizes the maximum-norm stability of BDF and finite element methods with respect to the boundary data, the discrete maximal $L^p$-regularity of BDF methods for parabolic equations and the Nevanlinna–Odeh multiplier technique combined with a time-dependent inner product motivated by a spectrum estimate of the linearized AC operator.