Adaptive Finite Element Method For Phase Field Crystal Model

Finite element method is widely used in scientific and engineering computing ow-ing to its complete mathematical theory and good adaptability to irregular geometries.Although a lot of research work has been done on finite element method,there are still many interesting problems to be further discussed.The standard priori error estimation of finite element method only gives the asymptotic relationship between mesh size and finite element error,but does not reflect the influence of mesh quality?such as element shape,size and mesh symmetry?on the approximation accuracy of finite element solution.In this paper,by means of element analysis,two com-putable quantities,G_eand G_v,were constructed to characterize the relationship between mesh quality and finite element error,so a more precise finite element error estimation was given.Based on the new error estimates,the classical superconvergence results were further derived.A lot of experiments showed that for isotropic and anisotropic meshes and problems,G_eand G_v could effectively estimate the approximation error of finite element numerical solutions.For mildly structured meshes and CVDT meshes,the finite element error|u-u_h|₁is equivalent to the interpolation error|u-u_I|₁.Using this equivalence,piecewise linear interpolation error expansion and Hessian recovery techniques,a H¹type posteriori error estimator is constructed.Numerical results verified the accuracy and validity of the error estimator.A simple and effective adaptive finite element method was developed for phase field crystal model.This method used scalar auxiliary variable?SAV?approach in time,which guarantees theoretically the energy dissipation property of numerical scheme.In order to capture phase interface better,L²norm of recovery gradient on element is used as adaptive indicator.Taking Landau-Brazovskii model as an example,Allen-Cahn type dynamic equation was used to simulate the mesoscale self-assembly in two-dimensional geometries.Numerical experiments showed that the method can accurately simulate the phase transition process and obtain standard ordered structures in general convex region.