The lumped mass FEM for a time-fractional cable equation

Mariam Al-Maskari, Samir Karaa

Research output: Contribution to journalArticle

1 Citation (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

Fingerprint

Cable
Error Estimates
Cables
Fractional
Finite Element Method
Finite element method
Mesh
Norm
Galerkin Finite Element Method
Piecewise Linear Function
Optimal Error Estimates
Fractional Derivative
Numerical Approximation
Quadrature
Difference Method
Euler
Convolution
Numerical Examples
Derivatives

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

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

In: Applied Numerical Mathematics, Vol. 132, 01.10.2018, p. 73-90.

Research output: Contribution to journalArticle

@article{d1bb2604110b41aabf0e35477bae6b4b,
title = "The lumped mass FEM for a time-fractional cable equation",
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.",
keywords = "Convolution quadrature, Error estimate, Laplace transform, Lumped mass FEM, Nonsmooth data, Time-fractional cable equation",
author = "Mariam Al-Maskari and Samir Karaa",
year = "2018",
month = "10",
day = "1",
doi = "10.1016/j.apnum.2018.05.012",
language = "English",
volume = "132",
pages = "73--90",
journal = "Applied Numerical Mathematics",
issn = "0168-9274",
publisher = "Elsevier",

}

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 -