The subdiffusion equation with a Caputo fractional derivative of order $\alpha \in (0,1)$ in time arises in a wide variety of practical applications, and it is often adopted to model anomalous subdiffusion processes in heterogeneous media. The L1 scheme is one of the most popular and successful numerical methods for discretizing the Caputo fractional derivative in time. The scheme was analysed earlier independently by Lin and Xu (2007, Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys., 225, 1533–1552) and Sun and Wu (2006, A fully discrete scheme for a diffusion wave system. Appl. Numer. Math., 56, 193–209), and an $O(\tau ^{2-\alpha })$ convergence rate was established, under the assumption that the solution is twice continuously differentiable in time. However, in view of the smoothing property of the subdiffusion equation, this regularity condition is restrictive, since it does not hold even for the homogeneous problem with a smooth initial data. In this work, we revisit the error analysis of the scheme, and establish an $O(\tau )$ convergence rate for both smooth and nonsmooth initial data. The analysis is valid for more general sectorial operators. In particular, the L1 scheme is applied to one-dimensional space-time fractional diffusion equations, which involves also a Riemann–Liouville derivative of order $\beta \in (\frac {3}{2},2)$ in space, and error estimates are provided for the fully discrete scheme. Numerical experiments are provided to verify the sharpness of the error estimates, and robustness of the scheme with respect to data regularity.

In this work, we consider boundary value problems involving either Caputo or Riemann-Liouville fractional derivatives of order $\alpha \in (1,2)$ on the unit interval $(0,1)$ . These fractional derivatives lead to nonsymmetric boundary value problems, which are investigated from a variational point of view. The variational problem for the Riemann-Liouville case is coercive on the space $H_0^{\alpha /2}(0,1)$ but the solutions are less regular, whereas that for the Caputo case involves different test and trial spaces. The numerical analysis of these problems requires the so-called shift theorems which show that the solutions of the variational problem are more regular. The regularity pickup enables one to establish convergence rates of the finite element approximations. The analytical theory is then applied to the Sturm-Liouville problem involving a fractional derivative in the leading term. Finally, extensive numerical results are presented to illustrate the error estimates for the source problem and eigenvalue problem.

We study the Rayleigh–Stokes problem for a generalized second-grade fluid which involves a Riemann–Liouville fractional derivative in time, and present an analysis of the problem in the continuous, space semidiscrete and fully discrete formulations. We establish the Sobolev regularity of the homogeneous problem for both smooth and nonsmooth initial data \(v\), including \(v\in L^2(\Omega )\). A space semidiscrete Galerkin scheme using continuous piecewise linear finite elements is developed, and optimal with respect to initial data regularity error estimates for the finite element approximations are derived. Further, two fully discrete schemes based on the backward Euler method and second-order backward difference method and the related convolution quadrature are developed, and optimal error estimates are derived for the fully discrete approximations for both smooth and nonsmooth initial data. Numerical results for one- and two-dimensional examples with smooth and nonsmooth initial data are presented to illustrate the efficiency of the method, and to verify the convergence theory.

We consider the initial-boundary value problem for an inhomogeneous time-fractional diffusion equation with a homogeneous Dirichlet boundary condition, a vanishing initial data and a nonsmooth right-hand side in a bounded convex polyhedral domain. We analyse two semidiscrete schemes based on the standard Galerkin and lumped mass finite element methods. Almost optimal error estimates are obtained for right-hand side data $$f(x,t)\in L^{\infty}(0,T;\dot H^{q}(\Omega))$$, − 1 < q ≤ 1, for both semidiscrete schemes. For the lumped mass method, the optimal L²(Ω)-norm error estimate requires symmetric meshes. Finally, two-dimensional numerical experiments are presented to verify our theoretical results.

In this work, a complete error analysis is presented for fully discrete solutions of the subdiffusion equation with a time-dependent diffusion coefficient, obtained by the Galerkin finite element method with conforming piecewise linear finite elements in space and backward Euler convolution quadrature in time. The regularity of the solutions of the subdiffusion model is proved for both nonsmooth initial data and incompatible source term. Optimal-order convergence of the numerical solutions is established using the proven solution regularity and a novel perturbation argument via freezing the diffusion coefficient at a fixed time. The analysis is supported by numerical experiments.