### 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 |
---|---|

Pages (from-to) | 73-90 |

Number of pages | 18 |

Journal | Applied Numerical Mathematics |

Volume | 132 |

DOIs | |

Publication status | Published - Oct 1 2018 |

### Fingerprint

### 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

### Cite this

*Applied Numerical Mathematics*,

*132*, 73-90. https://doi.org/10.1016/j.apnum.2018.05.012

**The lumped mass FEM for a time-fractional cable equation.** / Al-Maskari, Mariam; Karaa, Samir.

Research output: Contribution to journal › Article

*Applied Numerical Mathematics*, vol. 132, pp. 73-90. https://doi.org/10.1016/j.apnum.2018.05.012

}

TY - JOUR

T1 - The lumped mass FEM for a time-fractional cable equation

AU - Al-Maskari, Mariam

AU - Karaa, Samir

PY - 2018/10/1

Y1 - 2018/10/1

N2 - 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∈Hq(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.

AB - 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∈Hq(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.

KW - Convolution quadrature

KW - Error estimate

KW - Laplace transform

KW - Lumped mass FEM

KW - Nonsmooth data

KW - Time-fractional cable equation

UR - http://www.scopus.com/inward/record.url?scp=85047599109&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85047599109&partnerID=8YFLogxK

U2 - 10.1016/j.apnum.2018.05.012

DO - 10.1016/j.apnum.2018.05.012

M3 - Article

VL - 132

SP - 73

EP - 90

JO - Applied Numerical Mathematics

JF - Applied Numerical Mathematics

SN - 0168-9274

ER -