Several Kinds Of Finite Element Methods For Time Distributed-order Partial Differential Equations

Time distributed-order differential equations are often used to describe the complex processes in which diffusion index changing with time,such as accelerated sub-diffusion processes,etc.At present,they have played an important role in many fields and have be-come a hot research topic in the international academic community.But the distributed-order differential equations are difficult to be solved because of the complexity and non-locality of distributed-order operators.Thus numerical methods have been considered by scholars.Among many algorithms,the finite element method is concerned by scholars due to its stronger regional adaptability,more flexible meshing,lower smoothness require-ments and stronger versatility compared with other numerical methods.In this thesis,several kinds of finite element methods are considered,such as H1-Galerkin mixed finite element(GMFE)method,two-grid finite element method and alternating direction im-plicit(ADI)finite element method.The detailed contents are summarized in the following three parts:In the first part,an H1-Galerkin mixed finite element method with some second-order a schemes is used to numerically solve a nonlinear time distributed-order sub-diffusion equation.Some second-order a schemes are chosen to approximate the time direction Meanwhile,the mixed finite element method is developed to discretize the space direction The stability and optimal priori error estimates for both p and auxiliary function u are derived in detail.Finally,a numerical example is given to confirm our theoretical analysis It is easy to see that the numerical solution figures are consistent with the exact solution figures,which indicates that the H1-GFEM method is effective for solving nonlinear time distributed-order partial differential equationsIn the second part,a two-dimensional nonlinear time distributed-order fractional sub-diffusion equation is solved by a two-grid ADI finite element method based on weighted and shifted Grunwald difference(WSGD)operator.The time distributed-order derivative is discretized by the numerical quadrature formula combined with WSGD operator.Then the two-grid ADI finite element fully discrete scheme is arrived at.The stability and opti-mal error estimates with second-order convergence rate in spatial direction are obtained The storage space can be reduced and the computing efficiency can be improved in this method.Two numerical examples are provided to verify the theoretical analysis.The comparison of the numerical solution and the exact solution is made to demonstrate the efficiency of the numerical methodIn the third part,an ADI finite element method based on the shifted fractional trapezoidal rule(SFTR)is discussed for the first time to solve a nonlinear time distributed-order coupled sub-diffusion model.The SFTR combined with the numerical quadrature formula is used for the time distributed-order derivative.Then the ADI finite element fully discrete scheme is obtained.The stability and error estimates for both unknown functions u and v are derived.According to the theoretical analysis and numerical calculation results,the optimal convergence order of this method is achieved in the spatial direction This method can largely reduce the huge demand for storage and can greatly improve computing efficiency.