Error estimates for approximations of distributed order time fractional diffusion with nonsmooth data (Q261251)

The authors consider the numerical approximation of a distributed order subdiffusion equation, arising in the modeling of ultra-slow diffusion processes. A space semidiscrete scheme based on the Galerkin finite element method is developed along with optimal error estimates in \(L^2(\Omega)\) and \(H^1(\Omega)\) norms for both smooth and nonsmooth initial data. Two fully discrete schemes, based on the Laplace transform and convolution quadrature generated by the backward Euler method, respectively, are presented along with optimal \(L^2(\Omega)\) error estimates, which exhibits exponential convergence and first-order convergence in time, respectively. Several numerical experiments are provided to support the error estimates for both smooth and nonsmooth initial data.