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We develop a fast space-time finite difference method for space-time fractional diffusion equations by fully utilizing the mathematical structure of the scheme. A circulant block preconditioner is proposed to further reduce the computational costs. The method has optimal-order memory requirement and approximately linear computational complexity. The method is not lossy, as no compression of the underlying numerical scheme has been employed. Consequently, the method retains the stability, accuracy, and, in particular, the conservation property of the underlying numerical scheme. Numerical experiments are presented to show the efficiency and capacity of long time modelling of the new method.

Fractional partial differential equations (FPDEs) provide better modeling capabilities for challenging phenomena with long-range time memory and spatial interaction than integer-order PDE do. A conventional numerical discretization of space-time FPDEs requires O(N 2 + M N ) memory and O(M N 3 + M 2 N ) computational work, where N is the number of spatial freedoms per time step and M is the number of time steps.We develop a fast finite difference method (FDM) for space-time FPDE: (i) We utilize the Toeplitz-like structure of the coefficient matrix to develop a matrix-free preconditioned fast Krylov subspace iterative solver to invert the coefficient matrix at each time step. (ii) We utilize a divide-andconquer strategy, a recursive direct solver, to handle the temporal coupling of the numerical scheme. The fast method has an optimal memory requirement of O(M N ) and an approximately linear computational complexity of O(N M (log N + log 2 M )), without resorting to any lossy compression. Numerical experiments show the utility of the method.

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MSC: 49J20 65M15 65M25 65M60 Keywords:Characteristic finite element method Optimal control problems Convection-diffusion equations Pointwise inequality constraints A priori error estimates a b s t r a c t In this paper we analyze a characteristic finite element approximation of convex optimal control problems governed by linear convection-dominated diffusion equations with pointwise inequality constraints on the control variable, where the state and co-state variables are discretized by piecewise linear continuous functions and the control variable is approximated by either piecewise constant functions or piecewise linear discontinuous functions. A priori error estimates are derived for the state, co-state and the control. Numerical examples are given to show the efficiency of the characteristic finite element method.