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In this work, we consider the numerical solution of an initial boundary valueproblem for the distributed order time fractional diffusion equation. The modelarises in the mathematical modeling of ultra-slow diffusion processes observedin some physical problems, whose solution decays only logarithmically as thetime $t$ tends to infinity. We develop a space semidiscrete scheme based on thestandard Galerkin finite element method, and establish error estimates optimalwith respect to data regularity in $L^2(D)$ and $H^1(D)$ norms for both smoothand nonsmooth initial data. Further, we propose two fully discrete schemes,based on the Laplace transform and convolution quadrature generated by thebackward Euler method, respectively, and provide optimal convergence rates inthe $L^2(D)$ norm, which exhibits exponential convergence and first-orderconvergence in time, respectively. Extensive numerical experiments are providedto verify the error estimates for both smooth and nonsmooth initial data, andto examine the asymptotic behavior of the solution.

Error Estimates for Approximations of Distributed Order Time Fractional Diffusion with Nonsmooth Data