Finite element analysis of a new phase field model with $p$-Laplacian operator

Abstract

In this paper, we propose a new phase field model involving the $p$-Laplacian operator with $1<p\le 2$. This newly developed model can be used to simulate the sub-diffuse interface phenomena in the course of phase transitions during microstructure evolution. We also propose an energy-stable finite element approximation for the proposed nonlinear parabolic problem and rigorously prove that the finite element approximation is unconditionally energy stable. Moreover, we give the optimal error estimate and convergence rate with respect to the mesh sizes. Numerical examples are carried out to demonstrate the stability and accuracy of the proposed scheme, which is in accordance with the theoretical ones. Numerical results also indicate that the evolution of free energy is in consistency with the discrete dissipation theory. Moreover, the energy initially decreases sharply when the parameter $p$ becomes larger. Meanwhile, we can observe that the parameter $p$ affects the spatial patterns during phase transition. More precisely, the phase field value will be changed more slowly as well when the parameter $p$ becomes smaller.

Introduction

The phase-field model has become a powerful tool for simulating the microstructure evolution in various fields of science and engineering, such as solidification [10], solid-state phase transformation [14], grain growth [26], [43], eutectic growth [3], [23], dislocation dynamics [46], [50], crack propagation [36], martensitic transformation [4], [51] and vesicle membranes in biological applications [8], [25]. In the content of phase-field models, the microstructural evolution is represented by means of a set of functions that is continuous in space and time. Clearly, the set of phase-field variables should be able to capture important physics behind the phase transformation and microstructural evolution. Nowadays, the phase-field model has gained considerable attention in quantitative aspects of simulations thanks to the ever-improving computational techniques.

Inspired by the work in [49], we propose a novel strategy based on the phase transition mechanism and the gradient flow approach to reformulate a phase field model, which involving the $p$-Laplacian operator. Such a model might be applicable to predict sub-diffuse interface phenomena during phase transition in microstructure evolution. As a matter of fact, the $p$-Laplacian operator has been proven to be an effective means for applications in many branches of science and engineering, such as nonlinear diffusion and filtration [44], non-Newtonian fluids [34], power-law materials [5], elastic–plastic torsional creep [45], flows in porous media [11] and so on. In mathematics, the $p$-Laplacian problems defined on a bounded domain have been studied extensively and many interesting results have been published, for example, in [2], [9], [12], [13], [20], [22], [37], [40] and references therein. Due to the presence of nonlinearity, it is not possible to find analytical solutions of these $p$-Laplacian problems. Therefore, in practice, numerical simulation becomes a crucial means for studying the dynamics of these equations. In the past years, numerical approximations of the $p$-Laplacian equations have been extensively done in [6], [7], [19], [21], [39], [42]. For the case with $p=2$, our investigated model is reduced to the classical Allen–Cahn phase field equation. This equation was originally introduced by Allen–Cahn [1] to describe the motion of antiphase boundaries in crystalline solids. It is now widely used in many complicated moving interface problems in fluid dynamics, materials science, image processing and biology [10], [33], [41]. One can also find a large body of numerical simulations of the Allen–Cahn equation, see [29], [30], [35], [47], [55], [57] and references therein. Noting that, in the existing literature, there have been extensive works for various gradient flows involved with $p$-Laplacian ($p=4$), such as epitaxial thin film growth [17], [32], [52] and square phase field crystal models [18]. Feng et al. [32] proposed an energy stable backward differentiation formula method for the epitaxial thin film equation with slope selection. Huang et al. [38] developed a preconditioned steepest descent (PSD) solver for the regularized $p$-Laplacian problem. Feng et al. [31] proposed a preconditioned steepest descent methods for some nonlinear elliptic equations involving p-Laplacian terms. In addition, there are many works that have been extensively reported in the phase-field systems, referring to [15], [16], [27], [53], [54], [56], [58] and references therein. To the best of our knowledge, numerical method and convergence analysis for the phase field equation with the $p$-Laplacian operator are still not available in the existing literature.

Regarding the existing works closely related to phase field model, it is worth mentioning that we first give the derivation of the phase field model involving the $p$-Laplacian operator, which can help us to describe some interesting properties of microstructure evolution that have not been observed in the classical phase field equation. It is remarkable that, there are several challenges to compute the numerical solutions of the phase field equation involving the $p$-Laplacian operator. In addition, dealing with the nonlinear potential term is quite challenging, because of its highly nonlinear nature. Moreover, the new phase field model satisfies energy dissipation laws. It is specifically desired to design a numerical scheme capable of preserving the energy stable property at the discrete level. Therefore, we attempt to develop some efficient and effective numerical schemes for solving the generalized phase field model involving the $p$-Laplacian operator.

The remainder of this paper is organized as follows. In the next section, we firstly derive the new phase field equation with the $p$-Laplacian operator under investigation, and we also introduce some notations and preliminaries needed in our later analysis. In Section 3, we establish the semi-discrete scheme using the finite element approximation. Also, the optimal error estimate with respect to the mesh size $h$ shall be proved. Section 4 gives the energy stability and convergence results for the fully discrete scheme. In Section 5, we perform numerical simulations to justify the proposed schemes. Some concluding remarks are drawn in Section 6.