Optimal-order error estimates of finite element approximations to variable-order time-fractional diffusion equations without regularity assumptions of the true solutions

We study a fully discretized finite element approximation to variable-order Caputo and Riemann–Liouville time-fractional diffusion equations (tFDEs) in multiple space dimensions, which model solute transport in heterogeneous porous media and related applications. We prove error estimates for the proposed methods, which are discretized on an equidistant or graded temporal partition predetermined by the behavior of the variable order at the initial time, only under the regularity assumptions of the variable order, coefficients and the source term but without any regularity assumption of the true solutions. Roughly, we prove that the finite element approximations to variable-order Caputo tFDEs have optimal-order convergence rates on a uniform temporal partition. In contrast the finite element approximations to variable-order Riemann–Liouville tFDEs discretized on a uniform temporal partition achieve an optimal-order convergence rate if $\alpha (0)=\alpha ^{\prime}(0) = 0$ but a suboptimal-order convergence rate if $\alpha (0)>0$. In the latter case, optimal-order convergence rate can be proved by employing the graded temporal partition. We conduct numerical experiments to investigate the performance of the numerical methods and to verify the mathematical analysis.