Mixed FEM for Time-Fractional Diffusion Problems with Time-Dependent Coefficients

Abstract

In this paper, a mixed finite element method is applied in spatial directions while keeping time variable continuous to a class of time-fractional diffusion problems with time-dependent coefficients on a bounded convex polygonal domain. Based on an energy argument combined with a repeated application of an integral operator, optimal error estimates, which are optimal with respect to both approximation properties and regularity results, are derived for the semidiscrete problem with smooth as well as nonsmooth initial data. Specially, a priori error bounds for both primary and secondary variables in $$L^2$$-norm are established. Since the comparison between Fortin projection and the mixed Galerkin approximation of the secondary variable yields an improved rate of convergence, therefore, as a by-product, we derive $$L^p$$-estimates for the error in primary variable. Finally, some numerical experiments are conducted to confirm our theoretical findings.