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 is an element of H-q(Omega) boolean AND H0(1)(Omega), q = 1,2. For nonsmooth initial data, i.e., v is an element of L-2 (Omega), the optimal L-2(Omega)-norm error estimate requires an additional assumption on mesh, which is known to be satisfied for symmetric meshes. A quasi-optimal L-infinity(Omega)-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. (C) 2018 IMACS. Published by Elsevier B.V. All rights reserved.