Numerical schemes for the time-fractional mobile/immobile transport equation based on convolution quadrature

In this work, the numerical approximation of the time-fractional mobile/immobile transport equation is considered. We investigate the solution regularity for two types of the initial data regularities. By applying the continuous piecewise linear finite elements in space, we obtain the spatial semidiscrete Galerkin scheme and derive its error estimates. We then propose two finite element schemes for the equation by employing convolution quadrature based on the backward Euler and the second-order backward difference methods. The corresponding error estimates for the two schemes are also given. Numerical examples of the two-dimensional problems are shown to confirm the convergence theory results.