Introduction to discrete functional analysis techniques for the numerical study of diffusion equations with irregular data

Research output: Chapter in Book/Report/Conference proceedingConference PaperResearchpeer-review

Abstract

We give an introduction to discrete functional analysis techniques for stationary and transient diffusion equations. We show how these techniques are used to establish the convergence of various numerical schemes without assuming non-physical regularity of the data. For simplicity of exposure, we mostly consider linear elliptic equations, and we briefly explain how these techniques are adapted and extended to non-linear time-dependent meaningful models (Navier--Stokes equations, flows in porous media, etc.). These convergence techniques rely on Sobolev norms and discrete forms of functional analysis results. The discrete functional analysis tools presented here are versatile and applicable to a number of numerical methods and models.
Original languageEnglish
Title of host publicationProceedings of the 17th Biennial Computational Techniques and Applications Conference
PublisherCambridge University Press
Pages101-127
Number of pages27
Volume56
Publication statusPublished - 2015
EventComputational Techniques and Applications Conference 2014 - Australian National University, Canberra, Australia
Duration: 1 Dec 20143 Dec 2014
Conference number: 17th
http://maths.anu.edu.au/events/ctac-2014

Conference

ConferenceComputational Techniques and Applications Conference 2014
Abbreviated titleCTAC2014
CountryAustralia
CityCanberra
Period1/12/143/12/14
Internet address

Keywords

  • elliptic equations
  • parabolic equations
  • non-linear equations
  • convergence analysis
  • discrete functional analysis
  • compactness
  • non-smooth data

Cite this

Droniou, J. (2015). Introduction to discrete functional analysis techniques for the numerical study of diffusion equations with irregular data. In Proceedings of the 17th Biennial Computational Techniques and Applications Conference (Vol. 56, pp. 101-127). Cambridge University Press.
Droniou, Jerome. / Introduction to discrete functional analysis techniques for the numerical study of diffusion equations with irregular data. Proceedings of the 17th Biennial Computational Techniques and Applications Conference. Vol. 56 Cambridge University Press, 2015. pp. 101-127
@inproceedings{2d78d022817e4a90b62270f11ac7c420,
title = "Introduction to discrete functional analysis techniques for the numerical study of diffusion equations with irregular data",
abstract = "We give an introduction to discrete functional analysis techniques for stationary and transient diffusion equations. We show how these techniques are used to establish the convergence of various numerical schemes without assuming non-physical regularity of the data. For simplicity of exposure, we mostly consider linear elliptic equations, and we briefly explain how these techniques are adapted and extended to non-linear time-dependent meaningful models (Navier--Stokes equations, flows in porous media, etc.). These convergence techniques rely on Sobolev norms and discrete forms of functional analysis results. The discrete functional analysis tools presented here are versatile and applicable to a number of numerical methods and models.",
keywords = "elliptic equations, parabolic equations, non-linear equations, convergence analysis, discrete functional analysis, compactness, non-smooth data",
author = "Jerome Droniou",
year = "2015",
language = "English",
volume = "56",
pages = "101--127",
booktitle = "Proceedings of the 17th Biennial Computational Techniques and Applications Conference",
publisher = "Cambridge University Press",
address = "United Kingdom",

}

Droniou, J 2015, Introduction to discrete functional analysis techniques for the numerical study of diffusion equations with irregular data. in Proceedings of the 17th Biennial Computational Techniques and Applications Conference. vol. 56, Cambridge University Press, pp. 101-127, Computational Techniques and Applications Conference 2014, Canberra, Australia, 1/12/14.

Introduction to discrete functional analysis techniques for the numerical study of diffusion equations with irregular data. / Droniou, Jerome.

Proceedings of the 17th Biennial Computational Techniques and Applications Conference. Vol. 56 Cambridge University Press, 2015. p. 101-127.

Research output: Chapter in Book/Report/Conference proceedingConference PaperResearchpeer-review

TY - GEN

T1 - Introduction to discrete functional analysis techniques for the numerical study of diffusion equations with irregular data

AU - Droniou, Jerome

PY - 2015

Y1 - 2015

N2 - We give an introduction to discrete functional analysis techniques for stationary and transient diffusion equations. We show how these techniques are used to establish the convergence of various numerical schemes without assuming non-physical regularity of the data. For simplicity of exposure, we mostly consider linear elliptic equations, and we briefly explain how these techniques are adapted and extended to non-linear time-dependent meaningful models (Navier--Stokes equations, flows in porous media, etc.). These convergence techniques rely on Sobolev norms and discrete forms of functional analysis results. The discrete functional analysis tools presented here are versatile and applicable to a number of numerical methods and models.

AB - We give an introduction to discrete functional analysis techniques for stationary and transient diffusion equations. We show how these techniques are used to establish the convergence of various numerical schemes without assuming non-physical regularity of the data. For simplicity of exposure, we mostly consider linear elliptic equations, and we briefly explain how these techniques are adapted and extended to non-linear time-dependent meaningful models (Navier--Stokes equations, flows in porous media, etc.). These convergence techniques rely on Sobolev norms and discrete forms of functional analysis results. The discrete functional analysis tools presented here are versatile and applicable to a number of numerical methods and models.

KW - elliptic equations

KW - parabolic equations

KW - non-linear equations

KW - convergence analysis

KW - discrete functional analysis

KW - compactness

KW - non-smooth data

M3 - Conference Paper

VL - 56

SP - 101

EP - 127

BT - Proceedings of the 17th Biennial Computational Techniques and Applications Conference

PB - Cambridge University Press

ER -

Droniou J. Introduction to discrete functional analysis techniques for the numerical study of diffusion equations with irregular data. In Proceedings of the 17th Biennial Computational Techniques and Applications Conference. Vol. 56. Cambridge University Press. 2015. p. 101-127