Document Zbl 0562.76086

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W.</a>; <a href="/authors/morton.k-william" title="Author Profile">Morton, K. W.</a></div> <h2 class="title"> <strong>Approximate symmetrization and Petrov-Galerkin methods for diffusion- convection problems.</strong> <i>(English)</i> <a class="label nowrap" href="/0562.76086">Zbl 0562.76086</a> </h2> <div class="source"> <a href="/serials/64" title="Journal Profile">Comput. Methods Appl. Mech. Eng.</a> <a href="/?q=in%3A148284" title="Articles in this Issue">45, 97-122 (1984)</a>. </div> <div class="abstract">The authors study the following diffusion-convection problem: $-\nabla \cdot (a\nabla u-bu)=f$ in $\Omega \in {\mathbb{R}}^ d$, $(a>0$, $\nabla \cdot b=0)$; $u=g$ on $\Gamma_ D$, $\partial u/\partial n=0$ on $\Gamma_ N$, where the boundary $\Gamma_ D\cup \Gamma_ N$ of $\Omega$ is such that the Dirichlet section $\Gamma_ D\neq \{\emptyset \}$ includes all the inflow boundary, i.e., $b\cdot n\geq 0$ on $\Gamma_ N$, where $n=outward$ normal. By carefully introducing the concept of symmetrization, the authors derive general error bounds for a Petrov- Galerkin method, provide approximations based on the symmetric forms associated with the problem, they also address the problem of how to interpret a finite element approximation which attempts to achieve optimality in one of the symmetric forms and finally generate numerical examples for one- and two-dimensional cases.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A221">M.A.Ibiejugba</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A3896559">Cited in <strong>49</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A76Rxx" title="MSC2020">76Rxx</a> </td> <td class="space"> Diffusion and convection </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A76M99" title="MSC2020">76M99</a> </td> <td class="space"> Basic methods in fluid mechanics </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Adiffusion-convection+problem">diffusion-convection problem</a>; <a href="/?q=ut%3Asymmetrization">symmetrization</a>; <a href="/?q=ut%3Aerror+bounds">error bounds</a>; <a href="/?q=ut%3APetrov-+Galerkin+method">Petrov- Galerkin method</a>; <a href="/?q=ut%3Anumerical+examples">numerical examples</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 0562.76086" data-ciurl="/ci/03896559" data-biburl="/bibtex/03896559.bib" data-amsurl="/amsrefs/03896559.bib" data-xmlurl="/xml/03896559.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/03896559.pdf" title="Zbl 0562.76086 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1016/0045-7825(84)90152-X" aria-label="DOI for “Approximate symmetrization and Petrov-Galerkin methods for diffusion- convection problems”" title="10.1016/0045-7825(84)90152-X">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Ciarlet, P. G., The Finite Element Method for Elliptic Problems (1978), North-Holland: North-Holland Amsterdam · <a href="/0445.73043" class="nowrap">Zbl 0445.73043</a></td> </tr><tr> <td>[2]</td> <td class="space">Morton, K. W., Finite element methods for non-self-adjoint problems, (Turner, P. R., Proc. SERC Summer School. Proc. SERC Summer School, Lancaster, 1981. Proc. SERC Summer School. Proc. SERC Summer School, Lancaster, 1981, Lecture Notes in Mathematics 965 (1982), Springer-Verlag: Springer-Verlag Berlin), 113-148 · <a href="/0551.65075" class="nowrap">Zbl 0551.65075</a></td> </tr><tr> <td>[3]</td> <td class="space">Heinrich, J. C.; Zienkiewicz, O. C., The finite element method and ‘upwinding’ techniques in the numerical solution of convection dominated flow problems, (Hughes, T. J.R., Finite Element Methods for Convection Dominated Flows, AMD, Vol. 34 (1979), ASME: ASME New York), 105-136 · <a href="/0436.76062" class="nowrap">Zbl 0436.76062</a></td> </tr><tr> <td>[4]</td> <td class="space">(Hughes, T. J.R., Finite Element Methods for Convection Dominated Flows, AMD, Vol. 34 (1979), ASME: ASME New York)</td> </tr><tr> <td>[5]</td> <td class="space">Barrett, J. W.; Morton, K. W., Optimal finite element solutions to diffusion convection problems in one dimension, Internat. J. Numer. Meth. Engrg., 15, 1457-1474 (1980) · <a href="/0442.76069" class="nowrap">Zbl 0442.76069</a></td> </tr><tr> <td>[6]</td> <td class="space">Barrett, J. W.; Morton, K. W., Optimal Petrov-Galerkin methods through approximate symmetrization, IMA J. Numer. Anal., 1, 439-468 (1981) · <a href="/0471.65059" class="nowrap">Zbl 0471.65059</a></td> </tr><tr> <td>[7]</td> <td class="space">Babuska, I.; Aziz, A. K., Survey lectures on the mathematical foundations of the finite element method, (Aziz, A. K., The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (1972), Academic Press: Academic Press New York), 3-363</td> </tr><tr> <td>[8]</td> <td class="space">Hemker, P. W., A numerical study of stiff two-point boundary problems, (Ph.D. Thesis (1977), Mathematisch Centrum: Mathematisch Centrum Amsterdam) · <a href="/0426.65043" class="nowrap">Zbl 0426.65043</a></td> </tr><tr> <td>[9]</td> <td class="space">Allen, D.; Southwell, R., Relaxation methods applied to determining the motion, in two dimensions, of a viscous fluid past a fixed cylinder, Quart. J. Mech. Appl. Math., 8, 129-145 (1955) · <a href="/0064.19802" class="nowrap">Zbl 0064.19802</a></td> </tr><tr> <td>[10]</td> <td class="space">Christie, I.; Griffiths, D. F.; Mitchell, A. R.; Zienkiewicz, O. C., Finite element methods for second order differential equations with significant first derivatives, Internat. J. Numer. Meth. Engrg., 10, 1389-1396 (1976) · <a href="/0342.65065" class="nowrap">Zbl 0342.65065</a></td> </tr><tr> <td>[11]</td> <td class="space">Heinrich, J. C.; Huyakorn, P. S.; Mitchell, A. R.; Zienkiewicz, O. C., An upwind finite element scheme for two-dimensional convective transport equations, Internat. J. Numer. Meth. Engrg., 11, 131-143 (1977) · <a href="/0353.65065" class="nowrap">Zbl 0353.65065</a></td> </tr><tr> <td>[12]</td> <td class="space">Hughes, T. J.R.; Brooks, A., A multi-dimensional upwind scheme with no crosswind diffusion, (Hughes, T. J.R., Finite Element Methods for Convection Dominated Flows, AMD, Vol. 34 (1979), ASME: ASME New York), 19-35 · <a href="/0423.76067" class="nowrap">Zbl 0423.76067</a></td> </tr><tr> <td>[13]</td> <td class="space">Johnson, C.; Nävert, U., An analysis of some finite element methods for advection-diffusion problems, (Axelsson, O.; Frank, L. S.; van der Sluis, A., Conf. on Analytical and Numerical Approaches to Asymptotic Problems in Analysis (1981), North-Holland: North-Holland Amsterdam) · <a href="/0455.76081" class="nowrap">Zbl 0455.76081</a></td> </tr><tr> <td>[14]</td> <td class="space">Hughes, T. J.R.; Brooks, A., A theoretical framework for Petrov-Galerkin methods with discontinuous weighting functions: application to the streamline-upwind procedure, (Gallagher, R. H.; Norrie, D. H.; Oden, J. T.; Zienkiewicz, O. C., Finite Elements in Fluids, Vol. 4 (1982), Wiley: Wiley New York), 47-65</td> </tr><tr> <td>[15]</td> <td class="space">Scotney, B., Numerical experiments and error analysis for Petrov-Galerkin methods, (Numer. Anal. Rept. 11/82 (1982), University of Reading) · <a href="/0603.65071" class="nowrap">Zbl 0603.65071</a></td> </tr><tr> <td>[16]</td> <td class="space">Griffiths, D. F.; Mitchell, A. R., On generating upwind finite element methods, (Hughes, T. J.R., Finite Element Methods for Convection Dominated Flows, AMD, Vol. 34 (1979), ASME: ASME New York), 91-104 · <a href="/0423.76069" class="nowrap">Zbl 0423.76069</a></td> </tr><tr> <td>[17]</td> <td class="space">Nävert, U., A finite element method for convection-diffusion problems, Ph.D. Thesis (1982), Gothenberg</td> </tr><tr> <td>[18]</td> <td class="space">Reinhardt, H. J., A-posteriori error analysis and adaptive finite element methods for singularly perturbed convection-diffusion equations, Math. Methods. Appl. Sci. (1984), to appear. · <a href="/0489.65056" class="nowrap">Zbl 0489.65056</a></td> </tr><tr> <td>[19]</td> <td class="space">Barrett, J. W.; Morton, K. W., Optimal finite element approximation for diffusion-convection problems, (Whiteman, J. R., The Mathematics of Finite Elements and Applications Proc.. The Mathematics of Finite Elements and Applications Proc., MAFELAP 1981 (1982), Academic Press: Academic Press London), 403-411 · <a href="/0504.76096" class="nowrap">Zbl 0504.76096</a></td> </tr><tr> <td>[20]</td> <td class="space">Barrett, J. W.; Moore, G.; Morton, K. W., Optimal recovery and defect correction in the finite element method, (Numer. Anal. Rept. 11/83 (1983), University of Reading) · <a href="/0678.65054" class="nowrap">Zbl 0678.65054</a></td> </tr><tr> <td>[21]</td> <td class="space">Murray, W. L., Numerical schemes for diffusion-convection in decelerating flows, Nat. Engrg. Lab. Rept. (1978)</td> </tr><tr> <td>[22]</td> <td class="space">Hutton, A. G., The numerical representation of convection, (IAHR Working Group Meeting (1981)) · <a href="/0474.76037" class="nowrap">Zbl 0474.76037</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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