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IMA Journal of Numerical Analysis Advance Access published online on June 23, 2009
IMA Journal of Numerical Analysis, doi:10.1093/imanum/drp002
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© The author 2009. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

The convection–diffusion Petrov–Galerkin story

K. William Morton{dagger}

Oxford University Computing Laboratory, Oxford OX1 3QD, UK

{dagger} Email: morton{at}comlab.ox.ac.uk

Received on 30 April 2008. Revised on 26 November 2008.


  Abstract

The term ‘Petrov–Galerkin method’ is probably due to Ron Mitchell and his collaborators, and he was certainly very active in studying the application of finite-element methods to second-order partial differential equations ‘with significant first derivatives’. Our aim in the present paper is to trace links between such early methods and the more recent discontinuous Galerkin methods—not only the methods but also their analysis. Also as Mitchell, like the author, was initially steeped in finite-difference methods, we shall sometimes use their manipulation in our analysis.

Key Words: convection-diffusion; Petrov-Galerkin; Finite-volume; Discontinuous-Galerkin


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