Convergence analysis of a discontinuous Galerkin method for a sub-diffusion equation

William McLean; Kassem Mustapha

doi:10.1007/s11075-008-9258-8

Convergence analysis of a discontinuous Galerkin method for a sub-diffusion equation

William McLean^*
, Kassem Mustapha

^*Corresponding author for this work

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

92 Scopus citations

Abstract

We employ a piecewise-constant, discontinuous Galerkin method for the time discretization of a sub-diffusion equation. Denoting the maximum time step by k, we prove an a priori error bound of order k under realistic assumptions on the regularity of the solution. We also show that a spatial discretization using continuous, piecewise-linear finite elements leads to an additional error term of order h² max (1,logk^-1). Some simple numerical examples illustrate this convergence behaviour in practice.

Original language	English
Pages (from-to)	69-88
Number of pages	20
Journal	Numerical Algorithms
Volume	52
Issue number	1
DOIs	https://doi.org/10.1007/s11075-008-9258-8
State	Published - Aug 2009

Bibliographical note

Funding Information:
We thank the University of New South Wales for financial support provided by a Faculty Research Grant.

Keywords

Finite elements
Memory term
Non-uniform time steps

ASJC Scopus subject areas

Applied Mathematics

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10.1007/s11075-008-9258-8

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abstract = "We employ a piecewise-constant, discontinuous Galerkin method for the time discretization of a sub-diffusion equation. Denoting the maximum time step by k, we prove an a priori error bound of order k under realistic assumptions on the regularity of the solution. We also show that a spatial discretization using continuous, piecewise-linear finite elements leads to an additional error term of order h2 max (1,logk-1). Some simple numerical examples illustrate this convergence behaviour in practice.",

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AB - We employ a piecewise-constant, discontinuous Galerkin method for the time discretization of a sub-diffusion equation. Denoting the maximum time step by k, we prove an a priori error bound of order k under realistic assumptions on the regularity of the solution. We also show that a spatial discretization using continuous, piecewise-linear finite elements leads to an additional error term of order h2 max (1,logk-1). Some simple numerical examples illustrate this convergence behaviour in practice.

KW - Finite elements

KW - Memory term

KW - Non-uniform time steps

UR - https://www.scopus.com/pages/publications/70350618343

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Convergence analysis of a discontinuous Galerkin method for a sub-diffusion equation

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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