Efficient Finite Element Methods For Time Fractional Diffusion Equations

Fractional partial differential equation is an important nonlocal equation,which has been widely used in modeling in various fields of science and engineering,such as anomalous diffusion,complex materials,fractal reservoir,image processing,com-putational neuroscience and so on.This promotes the unprecedented development of numerical methods for fractional partial differential equations.However,fractional partial differential equations contain fractional calculus operators,which makes the equations have weak singularity,nonlocality and even space-time coupling,which brings great challenges to the design of numerical schemes.Although many re-searchers have made some progress,there are still many problems to be solved.Here,we mainly discuss the efficient finite element method of time fractional diffusion e-quation with different characteristics,the main contents are as follows:Firstly,the fast finite element method is studied to solve the nonlinear time-fractional diffusion equation.The L1 scheme is discretized in time direction and the standard finite element method is discretized in spatial direction,so the implicit discrete scheme is obtained.Then,through the fractional Gr?nwall inequality,we proved the error estimates of discrete finite element scheme in L~2(?)norm,L~4(?)norm and H~1(?)norm,and the error estimates of fully discrete scheme in L~2(?)norm.Furthermore,two fast two-grid algorithms based on finite element discretization are constructed to solve the implicit discrete scheme,and the error estimates of the semi-discrete two-grid algorithms under L~2(?)and H~1(?)norm and the error estimates of the fully discrete two-grid algorithm under L~2(?)norm are analyzed.Finally,numerical examples demonstrate the high efficiency and reliability of the fast finite element discrete scheme.Then,a fast mixed finite element method is considered for nonlinear time-fractional diffusion equations with diffusion coefficients.Due to the existence of diffusion coefficients in the equation,which may be very small in practical applica-tions,the expanded mixed finite element method is used to discretethe diffusion space,and the L1 scheme is used to approximate the time fractional derivative.Thus,the fully discrete scheme is obtained.We strictly prove the stability of the fully discrete scheme under L~2(?)norm.Then,by defining the mixed elliptic projection,we ana-lyze the error estimates of L~2(?)norm for the fully discrete scheme,and obtain the L~perror estimate.Similarly,we construct a two-grid algorithm based on expanded mixed finite element discretization,and give the stability analysis and error analysis of the algorithm under L~2(?)norm.Finally,numerical examples are given to verify the validity of the expanded hybrid finite element scheme for the minimal diffusion coefficient models and the reliability of the proposed two-grid algorithm,which shows that the two-grid algorithm is more cost-effective than the direct numerical method.Next,the immersion finite element method is considered of time fractional diffu-sion equation with discontinuous coefficients.In the case of considering the singularity of the solution,the discretization is carried out by the non-uniform L1 method in time fractional derivative,and in order to deal with the discontinuous coefficient,the dis-cretization is carried out by the immersed finite element method in space direction.Based on the regularity of the solution and theαrobustness results of construct-ing the convolution discretization sum,We analyzed the fully discrete scheme under L~2(?)norm and broken H~1(?)norm.Furthermore,based on the stability results of the fully discrete scheme,we deduce the error estimates for the fully discrete scheme under the L~2(?)norm and H~1(?)broken norm.Finally,some numerical examples are used to verify the error results of our analysis.Finally,the partially penalized immersion finite element method(PPIFE)is con-sidered for time fractional diffusion equations with discontinuous coefficients.Taking the local penalty immersion finite element discretization of the elliptic interface prob-lem as the starting point,and then according to the characteristics of the weak form and discrete scheme of the elliptic interface problem,the continuous spectral space and discrete spectral space of the elliptic interface operator are defined by using the expansion technique of the elliptic operator.Then,the representation of the solution of the original problem is obtained by using the Laplace transform,and the regularity of the solution is given.Then,based on these theoretical foundations,The pointwise-in-time error estimation of semidiscrete PPIFE scheme under smooth and nonsmooth initial values is analyzed.Finally,our theory is verified by numerical experiments.