Finite Element Numerical Simulation For Fractional Phase Field Problems

Phase field models,featured by the scalar phase function,can describe diffusion pro-cesses and development of the interfaces for different phase components,which are widely used to describe such important natural phenomena as multiphase hydrodynamics,crys-tal growth,thermodynamics and fracture mechanics.Classical phase field models can be derived from the variation for Ginzburg-Landau free energy functional with double-well potential,in which the most famous models are the fourth-order Cahn-Hilliard equation and second-order Allen-Cahn equation.In the derivation of the classical phase field model,the long-range interactions between particles were expressed as a spatial convolution which was used to describe the effect,of long-range forces between particles,was simplified as an integer differential operator for an easy mathematical deduction.Therefore,it is necessary to establish fractional phase field models and develop efficient numerical methods to recover the long-range forces between particles.This dissertation was aimed at constructing efficient Galerkin finite element sim-ulations for the fractional phase field models and establishing their rigorous numerical analysis.By doing in this manner,it is expected to provide a theoretical foundation for practical phase field simulations.1.Finite element method for fractional diffusion equation and identification of fractional-orderBased on perceptions that fractional Laplace operator in phase field model can be re-garded as a special fractional diffusion operator,we first consider a finite element method for fractional diffusion problem with variable-coefficients.Meanwhile,to comply with the requirement for the determination for the fractional-order in the field test,we design a finite element identification algorithm to determine the fractional-order appearing in the unsteady-state fractional diffusion equations.These preparations make us have a better understanding for the difficulties and key techniques during the followed numerical sim-ulation for fractional problems and even non-local problems,and thus the experiences accumulated can help us better to realize the numerical simulation for fractional phase field models.(1)Least-squares mixed finite element for the fractional diffusion equations with variable coefficients.Due to the impact of the variable coefficient,the coerciveness can not be guaranteed for a directly established Galerkin form,and consequently the solv-ability of Galerkin form and the induced finite element scheme can not be obtained too.As a remedy we convert,by introducing the diffusion flux as an intermediate variab le,the original equation to a system of a first-order equation with variable-coefficient and a fractional equation with constant-coefficient.Then,we propose a least-squares mixed Galerkin formulation,for which the solvability is confirmed and a new result concerning the regularity of the solution is proved.Further,we construct a least-squares mixed finite element scheme to solve the unknown and the adjoint diffusion flux separately,and hence,the computation cost can be effectively reduced.We also prove that the finite element solution possesses the optimal-order approximation to the exact solution and the adjoint diffusion flux.(2)Finite element-Armijo algorithm for the identification of the fractional-order.Compared with the second-order diffusion equation,we only know that the fractional-order in fractional diffusion equations exists in a known interval,for example,the interval(1,2).However,the fractional-order should be previously and accurately determined in applications.This fact requires that some algorithms and the related mathematical theo-ry to determine the fractional-order should be given.For this reason,we propose a finite element-Armijo algorithm to identify the fractional-order through the following steps:based on the theory of Fourier transform,we prove that fractional Riemann-Liouville derivative operators satisfy semigroup property in somesense,and the L2-norm satisfies strong weak left continuity and weak right continuity with respect to the fractional-order;Furthermore,the identification is reduced to an optimal problem of a.convex functional in L2-norm sense;By using the obtained semigroup properties,we prove that the solution set to fractional diffusion equations forms a,weakly closed subset in a Sobolev space,and the minimal functional is a weak lower semi-continuous functional defined on this weakly closed subset.Therefore,this minimal functional achieves its minimum on the weakly closed subset,thus the existence and uniqueness of the optimal solution obtained;Based on the above analysis,we design a finite element-Armijo algorithm for fractional-order identification by using common finite element space and provide numerical experiments to confirm the reliability of this algorithm.2.Finite element simulations and theoretical analysis for fractional phase field problemsThis part is focused on finitc clement simulations for one-dimensional fractial Cahn-Hilliard equation,and then extended this idea to two-dimensional fractional Allen-Cahn and Cahn-Hilliard phase field models.(1)Mixed finite element simulation and theoretical analysis for one-dimensional fractional Cahn-Hilliard equations.by introducing an intermediate variable,we split the original equation as a system of two fractional Laplace nonlinear systems and con-struct an equivalent mixed variational form.Using the backward Euler scheme for time deriative and mixed finite element method for spatial diseretization,we construct a convex-split mixed finite element scheme.Applying Brouwer's fixed point principle,we prove the solvability of the discrete system and demonstrate that the discrete solution p-reserves the energy dissipation and a new-energy equality under a new energy definition.We use discrete Ehrling inequality to prove convergence property of the scheme under energy norm.To reduce the computational cost that is caused by the non-sparse stiffness matrix generated from the fractional Laplace operator,we combine Toeplitz block struc-ture of stiffness matrix,the fast Fourier transform(FFT)and the classical bi-conjugate gradient(BiCG)to design a fast bi-conjugate gradient algorithm(FBiCG),which reduces the computation cost and the storage from O(M2)of CG algorithm to O(M log M)and O(M),respectively.Numerical experiments also show that the convex-split mixed finite element formulation not only possesses good convergence accuracy and computational efficiency,but also preserves the originally physical natures,such as energy dissipation law,energy equality law under a new energy definition,the coarsening process and the influences on the interfaces from the fractional-order s and r,the diffusion coefficient K and some other parameters.(2)Finite element method and theoretical analysis for two-dimensional fractional Allen-Cahn model.Under G alerkin framework,we construct a Galerkin variational form and the corresponding finite element scheme,and then prove the solvability,stability.energy dissipation and convergence of the finite element solution.Numeric al experiments verify the convergence of the scheme,the energy law and the coarsening process of each component.(3)Finite element,algorithm for two-dimensional Cahin-Hilliard model.We extend the ideas for one-dimensional Cahn-Hilliard model to its two-dimensional counterpart,and construct the convex split-mixed element scheme,and then prove the existence and uniqueness,energy law and convergence rates for the finite element solution.Numerical experiments also confirm that the proposed algorithm,other than the optimal approx-imation,preserves the originally physical natures,such as energy dissipation law,the coarsening process.It is necessary to point out that,compared with the one-dimensional problem,the difficulties of calculation for higher-dimensional fractional phase field prob-lems are significantly increased,of which the biggest one is calculation of multiple singular integrals in the formation of stiffness matrix.