An initial-boundary value problem of the form \(D_t^\alpha - \varepsilon ^2 D_x^2 u + b u = f\) is considered on the space-time domain \([0,1]\times [0,T]\) , with Dirichlet boundary and initial conditions, where \(D _t ^{\alpha }\) is a Caputo fractional derivative of order \(\alpha \in (0,1)\) and the singular perturbation parameter \(\varepsilon \ll 1\) is a positive constant. Bounds on the solution u and its derivatives are proved by means of a comparison principle with a careful selection of barrier functions; it is seen that u has a weak singularity at the initial time \(t = 0\) (caused by the fractional derivative) and also has layers (caused by the small parameter \(\varepsilon\) ) at the sides \(\{(x,t): x = 0\text { or }1, 0<t\le T\}\) of the space-time domain. The Caputo derivative \(D_t^\alpha\) is discretised by the L1 scheme on a graded temporal mesh, then at each time level the PDE is discretised by a piecewise linear finite element method on a Shishkin spatial mesh. Using our bounds on the derivatives of u, error estimates for the computed solution are derived in \(L^2\) , energy and balanced norms on [0, 1] for each t; these estimates are local in time and uniform in \(\varepsilon\) . Numerical experiments show the sharpness of our theoretical results.
