The lumped mass FEM for a time-fractional cable equation

We consider the numerical approximation of a time-fractional cable equation involving two Riemann–Liouville fractional derivatives. We investigate a semidiscrete scheme based on the lumped mass Galerkin finite element method (FEM), using piecewise linear functions. We establish optimal error estimates for smooth and middly smooth initial data, i.e., v∈H^q(Ω)∩H₀ ¹(Ω), q=1,2. For nonsmooth initial data, i.e., v∈L²(Ω), the optimal L²(Ω)-norm error estimate requires an additional assumption on mesh, which is known to be satisfied for symmetric meshes. A quasi-optimal L^∞(Ω)-norm error estimate is also obtained. Further, we analyze two fully discrete schemes using convolution quadrature in time based on the backward Euler and the second-order backward difference methods, and derive error estimates for smooth and nonsmooth data. Finally, we present several numerical examples to confirm our theoretical results.