Mixed-type Galerkin Variational Principle And Numerical Simulation For A Generalized Nonlocal Elastic Model Lueling Jia

In this paper,we discuss a generalized nonlocal elastic model defined as follows:where ?=(0,1),d~+>0 and d~->0 are the left and right diffusivity coefficients,is the source or sink term,0D??u(x)and xD1?u(x)represent the left and right Riemann-Liouville fractional derivative operators respectively.Since the integral and derivative appear in this model at the same time,it's natural to introduce an intermediate variable to solve this integral-differential equa-tion.We regard the fractional derivative part in the integrand as a new variable,then decouple the original system into an integral equation with order 1-? and a derivative equation with order ?,and therefore we can define its mixed-type Galerkin variational principle.As these two equations don't need to satisfy coupling relations and can be solved independently,we only require to verify the corresponding bilin-ear forms are coercive and continuous and we can prove the wellposedness of the mixed-type variational problem in the space H(1-?)/2(?)×H0?/2(?).In the discussion of the solvability of the fractional integral equation,we also obtain a solvable cri-terion for the Fredholm integral equation of the first kind in ?-(1-?)/2(?).Based on the mixed-type variational principle we define its mixed-type finite element scheme and prove that there exists a unique solution of this discreted procedure.For this numerical simulation,we use the properties of the interpolation operator and L2 projection operator to estimate the intermediate variable and the unknown under their energy norm respectively.Numerical experiments are induced to verify the precision of the scheme.Because of the nonlocality of the fractional operators,the stiff matrix in the linear equations obtained from the discreted scheme are usually dense.For a linear equations with N unknown variables,the memory requirement of the coefficient ma-trix is O(N2)and the computational cost of a direct solver(such as Gauss elimination method)is O(N3).When the index N increases,the complexity would make the calculation too time-consuming to possess the efficiency.As a result,it's necessary for us to design a fast solver for this problem.When we choose piesewise con-stant functions and piesewise linear functions to simulate the intermediate variable and the unknown respectively,we find that the corresponding coefficient matrices have,or partly have the Toeplitz structure.It's acknowledged that the storage for Toeplitz matrix can be reduced to O(N)and the computational cost for Toeplitz matrix-vector multiplication is O(N log N).So we could design a fast algorithm based on the conjugate gradient method to deal with this linear equations,with the storage to be O(N)and the cost to be O(N log N)pre iteration.For ill-conditioned matrix,a proper preconditioner would largely reduce the iterative number so as to further improve the efficiency of the algorithm.We conduct numerical experiments to substantiate the utility of the fast algorithm.