A wavelet-Galerkin scheme for the phase field model of microstructural evolution of materials

This paper presents a numerical scheme for solving the governing equations of the phase field model for microstructural evolution of materials. Grain-growth is used as an example but the scheme can be applied to any general phase field model. The numerical scheme is a Galerkin scheme using Daubechies wavelets as the trial and weight functions. Multi-level Daubechies wavelets with six vanishing moments are used. Using the wavelets as the trial and weight functions has the advantage of focusing the numerical calculation to where the field variable has high gradient, i.e. the interfacial boundaries. The numerical scheme therefore overcomes the problem of computational inefficiency in the phase field models. In this paper we first explain the wavelet-Galerkin scheme using a one-dimensional example for its simplicity and then extend the scheme to three-dimensional problems. By comparing the wavelet-Galerkin scheme with a classical finite difference scheme for both one and three-dimensional problems, the wavelet-Galerkin scheme is verified and its computational efficiency is demonstrated especially for three-dimensional problems.