The Allen-Cahn problem comprises of a singularly perturbed parabolic semi-linear parabolic partial differential equation (PDE) together with suitable initial and boundary conditions, viz.,
we assume that is a convex, polygonal () or polyhedral () domain of the Euclidean space , , , for sufficiently smooth initial condition and forcing function (precise regularity statements will be given below).
The problem (1) belongs to the class of the so-called phase field PDEs models for solidification of a pure material, originally introduced by Allen & Cahn  to describe the phase separation process of a binary alloy at a fixed temperature. The nonlinear function is the derivative of the classical double-well potential . Due to the nature of the non-linearity, the solution develops time-dependent interfaces , separating regions for which from regions where . The solution moves from one region to another within the, so-called, diffuse interfaces of length . For a recent comprehensive review of phase field models and their relationship to geometric flows, we refer to .
Realistically, should be orders of magnitude smaller than the physical domain of simulation. Therefore, the accurate and efficient numerical solution of such phase field models requires the resolution of the dynamic diffuse interfaces. This means that the discretisation parameters of any numerical method used should provide sufficient numerical resolution to approximate the interface evolution accurately. In the context of finite element methods, this is typically achieved via the use of very fine meshes in the vicinity of the interface region. In an effort to simulate at a tractable computational cost, especially for , it is essential to design adaptive algorithms which are able to dynamically modify the local mesh size.
A standard error analysis of finite element approximations of (1) leads to a priori estimates with unfavourable exponential dependence on . This is impractical even for moderately small interface length . The celebrated works [8, 10, 2]
showed that uniform bounds for the principal eigenvalue of the linearized Allen-Cahn spatial operator about the solutionare possible as long as the evolving interface is smooth (cf., (20) below). Such spectral estimates are used in the seminal work  whereby a priori error bounds with only polynomial dependence on for finite element methods have been proven, enabling also the proof of convergence to the sharp-interface limit. Moreover, assuming the validity of a spectral estimate about the exact solution , allowed the proof of the first conditional-type a posteriori error bounds for finite element methods approximating the Allen-Cahn problem in -norm, for which the condition depends only polynomially on ; this was presented in the influential works [18, 13].
This direction of research has taken a further leap forward with the seminal works [4, 6, 7], whereby the principle eigenvalue of the linearized spatial Allen-Cahn operator about the numerical solution is used instead, in an effort to arrive to fully computable a posteriori error estimates in - and -norms, the latter using the elliptic reconstruction framework [20, 19]. We also mention  whereby a posteriori error bounds in the -norms, are proven.
When the interface undergoes topological changes, however, e.g., when an interface collapses, unbounded velocities occur and the all-important principal eigenvalue can scale like on a time interval of length comparable to . This crucial observation, made in , showed that the principal eigenvalue can be assumed to be -integrable with respect to the time variable allowing, in turn, for robust conditional a posteriori error analysis under topological changes in - and -norms.
In a recent work , a priori bounds for the -norm error have been proved, which appear to deliver a rather favourable -polynomial dependence on the respective constant, noting that -norm is present in the stability of the spatial Allen-Cahn operator upon multiplication of (1) by and integration with respect to space and to time. An immediate question is whether proving conditional a posteriori error bounds in -norm norm can also improve the dependence of the condition on the interface length . Motivated by this, in this work, we prove conditional a posteriori error bounds for the -norm for a backward Euler-finite element method. The proof is valid under the hypothesis of the existence of a spectral estimate under topological changes in the spirit of . The argument uses a carefully constructed test function, in conjunction with a continuation argument and a new variant of the elliptic reconstruction introduced in . As a result of the method of proof, the new a posteriori error analysis provides also new - and -norm a posteriori error bounds which appear to, at least formally, be valid under less stringent smallness condition compared to results from the literature.
The remainder of this work is structured as follows. The model problem is introduced In Section 2. Section 3 include the definition of the numerical method along with the elliptic and time reconstructions needed for the proof of the main results. The key estimates and the main result are stated and proven in Section 4. Section 5 completes the derivation of fully computable error bounds by estimating the terms appearing in the residuals of the main results.
2. Model problem
We denote by , the standard Lebesgue spaces with corresponding norms . Let also is the th order of Sobolev space based on and , , along with the corresponding norms and , respectively. Set . We shall denote by the duality pairing between and , which becomes the standard inner product when the arguments are sufficiently smooth. The respective Bochner spaces are denoted by , endowed with the norms:
with a Banach space with norm .
We shall make extensive use of the classical Gagliardo-Nirenberg-Ladyzhenskaya inequalities (GNL) reading:
for all with , independent of . For later use, we also recall a basic algebraic estimate, often referred to as the Young’s inequality: for any , we have
for any and , for some independent of .
Let and . Then, for a.e. , we seek , such that
for all Integrating for , and integrating by parts the above becomes: find , such that
for all .
3. The fully discrete scheme and reconstructions
We shall first present a fully discrete scheme for the Allen-Cahn problem (1) by combining the lowest order discontinuous Galerkin time-stepping method with conforming finite elements in space. Further, we shall define suitable space and time reconstructions of the fully discrete scheme, which will be crucial for the proof of the a posteriori error bounds below.
Let . We partition the time interval into subintervals and we denote by , each time step.
Let also be a sequence of conforming and shape-regular triangulations of the domain , that are allowed to be modified between time steps. We define the meshsize function, , by , for . To each we associate the finite element space:
with denoting the -variate space of polynomials of degree at most . The whole theory presented below remains valid if box-type elements are used and respective polynomial spaces of degree on each variable.
We say that a set of triangulations is compatible when they are constructed by different refinements of the same (coarser) triangulation. Given two compatible triangulations and , we consider their finest common coarsening and set . The partial order relation among the triangulations implies that . Furthermore, we denote by the interior mesh skeleton of , and we define the sets and
Approximations will be subordinate to the time partition. A finite element space is specified on each time interval , . Then, we seek approximate solutions from the space
with denoting the space of constant polynomials over , having values in ; these functions are allowed to be discontinuous at the nodal points, but are taken to be continuous from the left.
3.2. Fully discrete scheme
For brevity, we set . The backward Euler-finite element method reads: for each , find , such that
for every , with and denoting the orthogonal -projection operator onto .
Let now defined by , for all , i.e., the discrete Laplacian. This allows for the strong representation of (6) as
Definition 3.1 (elliptic reconstruction).
For each we define the elliptic reconstruction to be the solution of the elliptic problem
Remark 3.2 (Galerkin orthogonality).
We observe that satisfies
This relation implies that is orthogonal to with respect to the Dirichlet inner product, a crucial property that allows to use a posteriori error bounds for elliptic problems to estimate various norms of from above; we refer to Section 5 for a detailed discussion.
Definition 3.3 (time reconstruction).
For , , we set
where the piecewise linear Lagrange basis function with .
The above definition implies that the time derivative of ,
is the discrete backward difference at .
4. A posteriori error estimates
We shall now use the reconstructions defined above, together with non-standard energy and continuation arguments and a spectral estimate for the linearized steady-state problem about the approximate solution , to arrive at a posteriori error bounds in the -, - and -norms.
4.1. Error relation
We begin by splitting the total error as follows:
In view of Remark 3.2, can be estimated by a posteriori error bounds for elliptic problems in various norms.
Also, satisfies an equation of the form (4) with a fully computable right-hand side that consists of and the problem data. To see this, (4) along with Definitions 3.1 and 3.3 and elementary manipulations lead to the following result.
Lemma 4.1 (error equation).
On , and for all , we have
Therefore, norms of can be estimated through PDE stability arguments; this will be performed below. Before doing so, however, we further estimate the term involving the elliptic reconstructions on the right-hand side from (14).
On , , we have
for all .
4.2. Energy argument
We begin by introducing some notation. We define
on each , , noting that ; for we adopt the convention that .
Moreover, for brevity, we also set
where , , , where is the constant of the Poincaré-Friedrichs inequality and as in (2).
Lemma 4.3 ().
Using Taylor’s theorem, we immediately deduce
Let , φορ , such that
Hypothesis implies that . Setting in (14), we have
Observing now the identities
elementary calculations yield
We shall further estimate each . We begin by splitting into
Applying Hölder, GNL for , Poincaré-Friedrichs and Young inequalities gives, respectively,
The Cauchy-Schwarz and Young inequalities also yield Likewise, we split as follows:
yielding the following bounds
From Lemma 4.2 and working as before, we have
Next, we split as follows:
and, using Hölder, Poincaré-Friedrichs and Young inequalities, we deduce