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)∈ L∞(0,T; Hq(Ω))|⁠, − 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.