Research On Finite Element Methods For The Time-fractional Schr(?)dinger Equation

The time-fractional Schr(?)dinger equation is a generalization of the classical Schr(?)-dinger equation,and it has been widely used in viscoelastic materials,fluid dynamics,plasma physics,nonlinear optics and so on.However,the explicit form of analytic solu-tion for this kind of equation is difficult to obtain,so it is of great theoretical significance and practical value to study its effective numerical methods.In this thesis,for the time fractional Schr(?)dinger equation,we transform it into its equivalent form by expressing its functions in the form of real and imaginary parts,and then study the efficient algo-rithm for the equivalent form based on its weakly singular solution.We discretize it by the finite element method in spacial direction.The L1 scheme and Alikhanov scheme on the Graded mesh are adopted to discretize the fractional derivatives in time respectively.Then the corresponding fully discrete schemes are obtained.We analyze the stability and optimal convergence accuracy of these schemes.At the same time,we give their cor-responding fast algorithms.Finally,numerical experiments are conducted to verify the theoretical results.The specific research content includes the following two parts:In the first part,we adopt the L1 scheme on the Graded grid and the finite ele-ment method to discretize the equivalent form in time and space direction,then the L1conforming finite element scheme and the L1 nonconforming finite element scheme are constructed.We analyze the stability of these two schemes.Then the optimal α-robust global error estimation inH~1norm for the L1 conforming finite element scheme,and the optimal local error estimation inL~2norm for these two formats are given respectively.At the same time,we give the Fast L1 algorithm to reduce the computing costs and the storage requirements.Finally,the optimal convergence order of the proposed methods and the high efficiency of the fast algorithms are verified by numerical examples.In the second part,based on the finite element method and the Alikhanov formula on Graded mesh,the fully discrete Alikhanov conforming finite element scheme is con-structed.Then we analyze its stability and obtain its optimal global error estimation inL~2norm.Furthermore,we derive the fast Alikhanov conforming finite element algorithm by applying the sum of exponential(SOE)method.Finally,numerical examples are given to verify the optimal convergence order and the advantage of the fast algorithm to improve the computational efficiency.