Numerical Methods For Some Nonlocal And Nonlinear Problems

With the development of science and engineering,more and more problems have the requirements for high performance numerical computations of complex systems,such as simulations and applications in mechanical engineering,material science,aerospace and satellite remote sensing and so on.More complex systems require more complex models including nonlocal and/or nonlinear operators.For example,anomalous diffusion phenomena which do not satisfy the Fickian law are ubiquitous in the natural sciences and social sciences.In fact,many complex dynamical systems often contain anomalous diffusion.The fractional differential operator with histor-ical dependence and nonlocal properties is an effective method to describe these complex systems.Usually,we use finite dimensional space to approximate infinite dimensional space of complex systems.Higher dimension brings better accuracy.But higher dimension leads to a large-scale and dense discretization system.To construct a fast numerical algorithm for a large-scale discretization problem or to reduce its scale is a great challenge.We conduct some research on the following aspects.First on the fractional differential equation model and its numerical algorithms and applications.The coexistence of immiscible liquid phases,liquid and vapor phases,vapor and solid phases,and liquid and solid phases are ubiquitous in nature and applications.The interfacial dynamics of immiscible fluids plays a very important role in the formation and phase change mechanism and the evolution of multi-phase systems,but is computationally a very difficult task.The development of computable math-ematical models for effectively describing multi-phase systems and corresponding effective and efficient numerical methods has been a major challenge in the study of multi-phase systems.On the other hand,fractional differential equation(FDE)models are emerging as powerful tools for modeling challenging phenomena involving long-time memory or long-range spatial interactions in many disciplines and appli-cations.FDE models also bring many new challenges in the theoretical analysis and numerical simulations.We present the following research:1.Fractional phase field model and its fast numerical algorithm.· The space-time fractional Allen-Cahn model and its numerical algorithm.Allen-Cahn model can be used to describe the physical processes such as so-lidification and crystallization.With the fractional Laplacian operator,it was shown that the fractional Allen-Cahn model exhibits a variety of modeling and numerical advantages over traditional integer-order Allen-Cahn model.In particular,the fractional model has flexibility of tunable sharpness.The fractional time derivative is introduced to describe the nonlocal memory ef-fect of the model.We develop an accurate and efficient numerical method which is fast and matrix-free with lower memory requirement for the space-time fractional Allen-Cahn model.The numerical scheme requires O(N log N)operations per iteration with O(N)memory requirement while a direct solver requires O(N3)computational operations with O(N2)memory requirement.So it is suitable for the large-scale system.Numerical experiments show the utility of the fractional phase-field model and the corresponding fast numer-ical method.They also show the effect of spatial and temporal fractional parameters on the tunable sharpness and delay behavior.· The space fractional Allen-Cahn Model and its second-order unconditional energy stable numerical schemes.Due to the complexity of the nonlinear po-tential function and the fractional differential operator in the model,there has not been a successful attempt to develop accurate and efficient schemes with a rigorously proven unconditionally energy stability property.We adopt In-variant Energy Quadratization approach,where some new auxiliary variables are introduced to enforce the mixed energy density as an invariant quadratic functional.The key point of Invariant Energy Quadratization method is that we reformulate the model system using the new variables to an equivalent sys-tem.The equivalent model is discretized in the new variables by treating the nonlinear terms semi-explicitly,which in turn produces a semi-discrete system of linear fractional Laplacian equations.The linear operator of semi-discrete system is shown to be symmetric and positive definite so that the equivalent system can be solved effectively and efficiently.First-order and second-order semi-discrete numerical schemes are presented and proven that they are ac-curate and unconditionally energy stable.Convergence test together with numerical simulations is presented after the semi-discrete schemes are fully discretized in space by collocation method to demonstrate the stability and the accuracy of the proposed schemes.2.Fractional differential equation model of shape-memory polymers and its numerical schemes.· A shape-memory polymer(SMP)is a polymeric material that is capable of memorizing its original shape,and can acquire a temporary shape upon defor-mation and returns to its permanent shape in response to an external stimulus such as a temperature change.As the material structure is heterogeneous in general,an integer-order differential equation model often fails to provide an appropriate description of the evolution process.Because SMPs can have sig-nificant changes of their shapes depending on whether an external stimulus temperature change exceeds their prescribed temperature,which in turn will significantly affect their microscopic network structure.Hence,a variable-order fractional-order differential equation model seems to be a more appro-priate model to describe the shape-memory behaviors of amorphous polymers.For the model,we present a numerical scheme for the forward problem with a known variable order.However,the variable order is generally unknown in practical problems,so we also study the inverse problem,i.e.,determining the variable order by known physical experimental data,and adaptive method is used to obtain solutions with acceptable accuracy.Some numerical experi-ments show the performance of the models.Second on the optimal control problem and its numerical algorithm.The mechanism of optimal control is to control the state(like temperature of a system,usually not easy to control directly)through the control variable,which drives the state through the state equation.Optimal control problems(OCPs)have been a lively and active mathematical field and cover various topics of practical problems such as time optimal control,feedback control,control of flow equations,optimal shape design and so on.We consider the adaptive finite element method(AFEM)for OCPs governed by the elliptic equation with pointwise constraints for control variable.· AFEM has many advantages.It aims at distributing more mesh nodes around the area where the singularities happen to reduce the scale and save the com-putational cost.Various types of reliable and efficient a posteriori error esti-mators which are used to detect the location of singularity and essential for AFEM have been developed in the last decades for OCPs.There exist some attempts to prove the convergence of AFEM for OCPs.The existing algo-rithms have two iterative stages with outer loops for adaptive iteration and inner loops for solving nonlinear problems iteratively.Convergence depends on both loops,which are coupled and affected by each other.In the existing algorithms,the error of inner loops isn't taken into account and huge compu-tational work cost in solving nonlinear problems iteratively.Using variational discretization concept to discretize the control variable and piecewise linear continuous finite elements to approximate state variable,we propose an algo-rithm consists of only one iterative stage,i.e.,the coupled nonlinear system is solved linearly.Based on the a posteriori error estimation for the OCPs with a reduction rate of data oscillation,a simple and efficient adaptive finite element method for optimal control problems is constructed with linear rate of conver-gence.Any prescribed error tolerance is thus achieved in a finite number of steps.Numerical experiments confirm our theoretical analysis.