## Abstract

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)∩H_{0}
^{1}(Ω), q=1,2. For nonsmooth initial data, i.e., v∈L^{2}(Ω), the optimal L^{2}(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.

Original language | English |
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Pages (from-to) | 73-90 |

Number of pages | 18 |

Journal | Applied Numerical Mathematics |

Volume | 132 |

DOIs | |

Publication status | Published - Oct 1 2018 |

## Keywords

- Convolution quadrature
- Error estimate
- Laplace transform
- Lumped mass FEM
- Nonsmooth data
- Time-fractional cable equation

## ASJC Scopus subject areas

- Numerical Analysis
- Computational Mathematics
- Applied Mathematics