Partial Differential Equations - Modelling and ... - ResearchGate
Partial Differential Equations - Modelling and ... - ResearchGate Partial Differential Equations - Modelling and ... - ResearchGate
XVI List of Contributors Igor Sazanov Civil and Computational Engineering Centre University of Wales-Swansea Swansea SA2 8PP Wales UK i.sazonov@swansea.ac.uk Danny C. Sorensen Rice University Department of Computational & Applied Mathematics Houston, TX, 77251-1892 USA sorensen@rice.edu Allen M. Tesdall Fields Institute Toronto,ON M5T 3J1 and Department of Mathematics University of Houston Houston, TX 77204 USA atesdall@fields.utoronto.ca Dan Tiba Romanian Academy Institute of Mathematics P.O. Box 1-764 RO-014700 Bucharest Romania dan.tiba@imar.ro Jari Toivanen Department of Mathematical Information Technology P.O. Box 35 (Agora) FI-40014 University of Jyväskylä, Finland Jari.Toivanen@mit.jyu.fi Nigel P. Weatherill Civil and Computational Engineering Centre University of Wales-Swansea Swansea SA2 8PP Wales UK N.P.Weatherill@swansea.ac.uk Mary F. Wheeler Institute for Computational Engineering & Sciences (ICES) University of Texas at Austin Austin, TX 78712 USA mfw@ices.utexas.edu
Discontinuous Galerkin Methods Vivette Girault 1 and Mary F. Wheeler 2 1 Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris VI, FR-75252 Paris cedex 05, France girault@ann.jussieu.fr 2 Institute for Computational Engineering and Sciences (ICES), University of Texas at Austin, Austin, TX 78712, USA mfw@ices.utexas.edu Summary. In this article, we describe some simple and commonly used discontinuous Galerkin methods for elliptic, Stokes and convection-diffusion problems. We illustrate these methods by numerical experiments. 1 Introduction and Preliminaries Discontinuous Galerkin (DG) methods use discontinuous piece-wise polynomial spaces to approximate the solution of PDE’s in variational form. The concept of discontinuous space approximations was introduced in the early 70’s, probably starting with the work of Nitsche [Nit71] in 1971 on domain decomposition and followed by a number of important contributions such as the work of Babu˘ska and Zlamal [BZ73], Crouzeix and Raviart [CR73], Rachford and Wheeler [RW74], Oden and Wellford [OW75], Douglas and Dupont [DD76], Baker [Bak77], Wheeler [Whe78], Arnold [Arn79, Arn82] and Wheeler and Darlow [WD80]. Afterward, interest in DG methods for elliptic problems declined probably because computing facilities at that time were not sufficient to solve efficiently such schemes. By the end of the 90’s, the thesis of Baumann [Bau97] and the spectacular increase in computing power, triggered a renewal of interest in discontinuous Galerkin methods for elliptic and parabolic problems. The work of Baumann was followed by numerous publications such as Oden, Babu˘ska and Baumann [OBB98], Baumann and Oden [BO99], Rivière et al. [RWG99, RWG01], Rivière [Riv00], Arnold et al. [ABCM02], among many others. Research on DG methods is now a very active field. In the meantime, discontinuous methods were applied extensively to hyperbolic problems [Bey94, BOP96]. One of the first is the upwind scheme introduced by Reed and Hill in their report [RH73] on neutron transport in 1973. The first numerical analysis was done by Lesaint and Raviart [LR74] in 1974 for the transport equation and by Girault and Raviart [GR79] in 1982 for the Navier–Stokes equations. We refer to the books by Pironneau [Pir89] and by Girault and Raviart [GR86] for a thorough study of this upwind scheme.
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Discontinuous Galerkin Methods<br />
Vivette Girault 1 <strong>and</strong> Mary F. Wheeler 2<br />
1 Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris VI,<br />
FR-75252 Paris cedex 05, France girault@ann.jussieu.fr<br />
2 Institute for Computational Engineering <strong>and</strong> Sciences (ICES),<br />
University of Texas at Austin, Austin, TX 78712, USA mfw@ices.utexas.edu<br />
Summary. In this article, we describe some simple <strong>and</strong> commonly used discontinuous<br />
Galerkin methods for elliptic, Stokes <strong>and</strong> convection-diffusion problems. We<br />
illustrate these methods by numerical experiments.<br />
1 Introduction <strong>and</strong> Preliminaries<br />
Discontinuous Galerkin (DG) methods use discontinuous piece-wise polynomial<br />
spaces to approximate the solution of PDE’s in variational form. The<br />
concept of discontinuous space approximations was introduced in the early<br />
70’s, probably starting with the work of Nitsche [Nit71] in 1971 on domain<br />
decomposition <strong>and</strong> followed by a number of important contributions such<br />
as the work of Babu˘ska <strong>and</strong> Zlamal [BZ73], Crouzeix <strong>and</strong> Raviart [CR73],<br />
Rachford <strong>and</strong> Wheeler [RW74], Oden <strong>and</strong> Wellford [OW75], Douglas <strong>and</strong><br />
Dupont [DD76], Baker [Bak77], Wheeler [Whe78], Arnold [Arn79, Arn82] <strong>and</strong><br />
Wheeler <strong>and</strong> Darlow [WD80]. Afterward, interest in DG methods for elliptic<br />
problems declined probably because computing facilities at that time were not<br />
sufficient to solve efficiently such schemes. By the end of the 90’s, the thesis of<br />
Baumann [Bau97] <strong>and</strong> the spectacular increase in computing power, triggered<br />
a renewal of interest in discontinuous Galerkin methods for elliptic <strong>and</strong> parabolic<br />
problems. The work of Baumann was followed by numerous publications<br />
such as Oden, Babu˘ska <strong>and</strong> Baumann [OBB98], Baumann <strong>and</strong> Oden [BO99],<br />
Rivière et al. [RWG99, RWG01], Rivière [Riv00], Arnold et al. [ABCM02],<br />
among many others. Research on DG methods is now a very active field.<br />
In the meantime, discontinuous methods were applied extensively to hyperbolic<br />
problems [Bey94, BOP96]. One of the first is the upwind scheme<br />
introduced by Reed <strong>and</strong> Hill in their report [RH73] on neutron transport in<br />
1973. The first numerical analysis was done by Lesaint <strong>and</strong> Raviart [LR74] in<br />
1974 for the transport equation <strong>and</strong> by Girault <strong>and</strong> Raviart [GR79] in 1982 for<br />
the Navier–Stokes equations. We refer to the books by Pironneau [Pir89] <strong>and</strong><br />
by Girault <strong>and</strong> Raviart [GR86] for a thorough study of this upwind scheme.
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