Error estimates for fully discrete BDF finite element approximations of the Allen–Cahn equation

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.

Analysis and approximations of an optimal control problem for the Allen-Cahn equation

“…The choice of the dG(0) method is due to the low regularity imposed by the optimal control setting. Other approaches for discretization of the Allen-Cahn can be found in [1,2] (BDF and extrapolated Runge-Kutta methods via an auxialiary variable formulation), [32] (symmetric interior penalty discontinuous Galerkin methods), [38,41] (scalar auxiliary variable / Crank-Nickolson scheme) [42,46] (second order semi-implicit scheme).…”

“…For the numerical analysis of the uncontrolled Allen-Cahn equation, i.e. for u = 0, this difficulty was first circumvented in the seminal works of [33], [36], [31], [7], [6], where error estimates (a-priori and a-posteriori) were established for the homogeneous Allen-Cahn equation with constants that dependent polynomially upon 1 ǫ based on suitable discrete approximation of the spectral estimate and a nonstandard continuation argument of the form of a nonlinear Gronwall Lemma. Unfortunately, such analysis typically requires regularity assumptions on the state (as well as to its fullydiscrete counterpart) that are not available within the optimal control context.…”