The lumped mass FEM for a time-fractional cable equation

Mariam Al-Maskari, Samir Karaa

Research output: Contribution to journalArticle

2 Citations (Scopus)

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∈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.

Original languageEnglish
Pages (from-to)73-90
Number of pages18
JournalApplied Numerical Mathematics
Volume132
DOIs
Publication statusPublished - Oct 1 2018

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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

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