Time-Space Adaptive Finite Element Method For Phase Field Equations

In this thesis,we focus on time-space adaptive finite element method for phase field equations.As an important physical model in computational mathematics,the phase field equations are essentially one of nonlinear partial differential equations.As the presence of the small parameter?,the phase field equations show the phenomenon of the boundary layer.In order to resolve the thin transition region,it is necessary to use very fine meshes so the spatial mesh size and time step size have properly relate to the interface width ?.To save the computational cost,it is natural to use adaptive mesh,rather than uniform mesh.As far as adaptation is concerned,good error indicators,which can be properly derived by theory,play a vital role.Gradient recovery,as one of the post-processing techniques for adaptive finite method,has a lot of advantages,such as improving the quality of the approximation,constructing a posteriori error estimators in adaptive computation,superconvergent property and so on.The SCR(Superconvergent Cluster Recovery)is such a gradient reconstruction method,thus the adaptive finite element method in this paper is mainly in view of the SCR based a posterior error estimation.We consider the adaptive finite element method for the Allen-Cahn equation.The adaptive method is based on a second order accurate unconditionally energy stable adaptive finite element scheme and a timespace adaptive algorithm is proposed for numerical approximation of the Allen-Cahn equation.For the error control,we take the difference of numerical approximations between two time steps as the time discretization error estimator and the difference between the numerical gradient and the recovered gradient by the superconvergent cluster recovery method as the space discretization error estimator to control the mesh refinement and coarsening.Numerical experiments are presented to illustrate the reliability and efficiency of the proposed SCR-based error estimator and the corresponding adaptive algorithm.The extension of the proposed adaptive algorithm to the Cahn-Hilliard equation is also discussed.In addition,we also derive recovery type a posteriori error estimation for the Allen-Cahn equation by using Crank-Nicolson finite element method.The derivation of error is based on elliptic reconstruction technique,which involves the splitting of the error into two parts:an elliptic error and a parabolic error.A time-space adaptive algorithm is proposed by using the recovery type a posteriori error estimator as the error indicators for the Allen-Cahn equation.Numerical experiments are presented to illustrate the reliability and efficiency of the derived a posteriori error estimator and the corresponding adaptive algorithm.We also apply the time-space adaptive finite element method to the Cahn-Hilliard-Navier-Stokes model.The adaptive method is based on a linear,decoupled scheme.A priori error analysis for fully discrete scheme is carried out to solve the Cahn-Hilliard-Navier-Stokes equations.An unconditionally energy stable discrete law for the modified energy is also shown for the fully discrete scheme.And a superconvergent cluster recovery based a posteriori error estimations are constructed for both the phase field variable and velocity field function,respectively.Based on the proposed space and time discretization error estimators,a timespace adaptive algorithm is designed for numerical approximation of the Cahn-Hilliard-Navier-Stokes equations.Numerical experiments are presented to illustrate the reliability and efficiency of the proposed error estimators and the corresponding adaptive algorithm.