Error analysis of semidiscrete finite element methods for inhomogeneous time-fractional diffusion

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) is an element of L-infinity(0,T;(H)over bar(q)(Omega)), -1 < q <= 1, for both semidiscrete schemes. For the lumped mass method, the optimal L-2(Omega)-norm error estimate requires symmetric meshes. Finally, two-dimensional numerical experiments are presented to verify our theoretical results.