Two temporal second-order H1-Galerkin mixed finite element schemes for distributed-order fractional sub-diffusion equations

Abstract

In this paper, a special point is found for the interpolation approximation of the distributed order fractional derivatives to achieve at least second-order accuracy. Then, two H1-Galerkin mixed finite element schemes combined with the higher accurate interpolation approximation are introduced and analyzed to solve the distributed order fractional sub-diffusion equations. The stable results, which just depend on initial value and source item, are derived. Some a priori estimates with optimal order of convergence both for the unknown function and its flux are established rigorously. It is shown that the H1-Galerkin mixed finite element approximations have the same rates of the convergence as in the classical mixed finite element method, but without LBB consistency condition and quasiuniformity requirement on the finite element mesh. Finally, some numerical experiments are presented to show the efficiency and accuracy of H1-Galerkin mixed finite element schemes.