Finite volume element method for two-dimensional fractional subdiffusion problems

Samir Karaa, Kassem Mustapha, Amiya K. Pani

Research output: Contribution to journalArticle

17 Citations (Scopus)

Abstract

In this article, a semidiscrete spatial finite volume (FV) method is proposed and analyzed for approximating solutions of anomalous subdiffusion equations involving a temporal fractional derivative of order α ? (0, 1) in a two-dimensional convex domain. An optimal error estimate in the L∞ (L2)-norm is shown to hold. A superconvergence result is proved, and as a consequence, it is proved that a quasi-optimal order of convergence in the L∞ (L∞ )-norm holds. We also consider a fully discrete scheme that employs a FV method in space and a piecewise linear discontinuous Galerkin method to discretize in time. It is further shown that the convergence rate is of order h2 + k1+α, where h denotes the spatial discretization parameter and k represents the temporal discretization parameter. Numerical experiments indicate optimal convergence rates in both time and space, and also illustrate that the imposed regularity assumptions are pessimistic.

Original languageEnglish
Pages (from-to)945-964
Number of pages20
JournalIMA Journal of Numerical Analysis
Volume37
Issue number2
DOIs
Publication statusPublished - Apr 1 2017

Fingerprint

Finite Volume Element Method
Subdiffusion
Finite volume method
Finite Volume Method
Fractional
Discretization
Norm
Optimal Convergence Rate
Optimal Error Estimates
Superconvergence
Discontinuous Galerkin Method
Convex Domain
Order of Convergence
Fractional Derivative
Galerkin methods
Piecewise Linear
Anomalous
Convergence Rate
Regularity
Numerical Experiment

Keywords

  • discontinuous Galerkin method
  • error analysis.
  • finite volume element
  • fractional diffusion

ASJC Scopus subject areas

  • Mathematics(all)
  • Computational Mathematics
  • Applied Mathematics

Cite this

Finite volume element method for two-dimensional fractional subdiffusion problems. / Karaa, Samir; Mustapha, Kassem; Pani, Amiya K.

In: IMA Journal of Numerical Analysis, Vol. 37, No. 2, 01.04.2017, p. 945-964.

Research output: Contribution to journalArticle

@article{80023c1942de407894e46578605db8b3,
title = "Finite volume element method for two-dimensional fractional subdiffusion problems",
abstract = "In this article, a semidiscrete spatial finite volume (FV) method is proposed and analyzed for approximating solutions of anomalous subdiffusion equations involving a temporal fractional derivative of order α ? (0, 1) in a two-dimensional convex domain. An optimal error estimate in the L∞ (L2)-norm is shown to hold. A superconvergence result is proved, and as a consequence, it is proved that a quasi-optimal order of convergence in the L∞ (L∞ )-norm holds. We also consider a fully discrete scheme that employs a FV method in space and a piecewise linear discontinuous Galerkin method to discretize in time. It is further shown that the convergence rate is of order h2 + k1+α, where h denotes the spatial discretization parameter and k represents the temporal discretization parameter. Numerical experiments indicate optimal convergence rates in both time and space, and also illustrate that the imposed regularity assumptions are pessimistic.",
keywords = "discontinuous Galerkin method, error analysis., finite volume element, fractional diffusion",
author = "Samir Karaa and Kassem Mustapha and Pani, {Amiya K.}",
year = "2017",
month = "4",
day = "1",
doi = "10.1093/imanum/drw010",
language = "English",
volume = "37",
pages = "945--964",
journal = "IMA Journal of Numerical Analysis",
issn = "0272-4979",
publisher = "Oxford University Press",
number = "2",

}

TY - JOUR

T1 - Finite volume element method for two-dimensional fractional subdiffusion problems

AU - Karaa, Samir

AU - Mustapha, Kassem

AU - Pani, Amiya K.

PY - 2017/4/1

Y1 - 2017/4/1

N2 - In this article, a semidiscrete spatial finite volume (FV) method is proposed and analyzed for approximating solutions of anomalous subdiffusion equations involving a temporal fractional derivative of order α ? (0, 1) in a two-dimensional convex domain. An optimal error estimate in the L∞ (L2)-norm is shown to hold. A superconvergence result is proved, and as a consequence, it is proved that a quasi-optimal order of convergence in the L∞ (L∞ )-norm holds. We also consider a fully discrete scheme that employs a FV method in space and a piecewise linear discontinuous Galerkin method to discretize in time. It is further shown that the convergence rate is of order h2 + k1+α, where h denotes the spatial discretization parameter and k represents the temporal discretization parameter. Numerical experiments indicate optimal convergence rates in both time and space, and also illustrate that the imposed regularity assumptions are pessimistic.

AB - In this article, a semidiscrete spatial finite volume (FV) method is proposed and analyzed for approximating solutions of anomalous subdiffusion equations involving a temporal fractional derivative of order α ? (0, 1) in a two-dimensional convex domain. An optimal error estimate in the L∞ (L2)-norm is shown to hold. A superconvergence result is proved, and as a consequence, it is proved that a quasi-optimal order of convergence in the L∞ (L∞ )-norm holds. We also consider a fully discrete scheme that employs a FV method in space and a piecewise linear discontinuous Galerkin method to discretize in time. It is further shown that the convergence rate is of order h2 + k1+α, where h denotes the spatial discretization parameter and k represents the temporal discretization parameter. Numerical experiments indicate optimal convergence rates in both time and space, and also illustrate that the imposed regularity assumptions are pessimistic.

KW - discontinuous Galerkin method

KW - error analysis.

KW - finite volume element

KW - fractional diffusion

UR - http://www.scopus.com/inward/record.url?scp=85019158859&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85019158859&partnerID=8YFLogxK

U2 - 10.1093/imanum/drw010

DO - 10.1093/imanum/drw010

M3 - Article

VL - 37

SP - 945

EP - 964

JO - IMA Journal of Numerical Analysis

JF - IMA Journal of Numerical Analysis

SN - 0272-4979

IS - 2

ER -