Document Zbl 0883.65088

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I: Methodology and convergence.</strong> <i>(English)</i> <a class="label nowrap" href="/0883.65088">Zbl 0883.65088</a> </h2> <div class="source"> <a href="/serials/64" title="Journal Profile">Comput. Methods Appl. Mech. Eng.</a> <a href="/?q=in%3A43539" title="Articles in this Issue">143, No. 1-2, 113-154 (1997)</a>. </div> <div class="abstract">Summary: This paper formulates the moving least-square interpolation scheme in a framework of the so-called moving least-square reproducing kernel (MLSRK) representation. In this study, the procedure of constructing moving least-square interpolation function is facilitated by using the notion of reproducing kernel formulation, which, as a generalization of the early discrete approach, establishes a continuous basis for a partition of unity. This new formulation possesses the quality of simplicity, and it is easy to implement. Moreover, the reproducing kernel formula proposed is not only able to reproduce any $m$th order polynomial exactly on an irregular particle distribution, but also serves as a projection operator that can approximate any smooth function globally with an optimal accuracy.<br class="zbmathjax-paragraph">In this contribution, a generic $m$-consistency relation has been found, which is the essential property of the MLSRK approximation. An interpolation error estimate is given to assess the convergence rate of the approximation. It is shown that for a sufficiently smooth function the interpolant expansion in terms of sampled values will converge to the original function in the Sobolev norms. As a meshless method, the convergence rate is measured by a new control variable – dilation parameter $\varrho$ of the window function, instead of the mesh size $h$ as usually done in the finite element analysis. To illustrate the procedure, convergence has been shown for the numerical solution of the second-order elliptic differential equations in a Galerkin procedure invoked with this interpolant. In the numerical example, a two-point boundary problem is solved by using the method, and an optimal convergence rate is observed with respect to various norms.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A1087063">Cited in <strong>1</strong> Review</a></div><div class="clear"><a href="/?q=rf%3A1087063">Cited in <strong>229</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A65N30" title="MSC2020">65N30</a> </td> <td class="space"> Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35J25" title="MSC2020">35J25</a> </td> <td class="space"> Boundary value problems for second-order elliptic equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65N12" title="MSC2020">65N12</a> </td> <td class="space"> Stability and convergence of numerical methods for boundary value problems involving PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65N15" title="MSC2020">65N15</a> </td> <td class="space"> Error bounds for boundary value problems involving PDEs </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3AGalerkin+method">Galerkin method</a>; <a href="/?q=ut%3Amoving+least-square+reproducing+kernel+method">moving least-square reproducing kernel method</a>; <a href="/?q=ut%3Aconsistency">consistency</a>; <a href="/?q=ut%3Ainterpolation">interpolation</a>; <a href="/?q=ut%3Aerror+estimate">error estimate</a>; <a href="/?q=ut%3Aconvergence">convergence</a>; <a href="/?q=ut%3Afinite+element+analysis">finite element analysis</a>; <a href="/?q=ut%3Anumerical+example">numerical example</a></div><div class="keywords"> <h3>Citations:</h3><a href="/0883.65089">Zbl 0883.65089</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 0883.65088" data-ciurl="/ci/01087063" data-biburl="/bibtex/01087063.bib" data-amsurl="/amsrefs/01087063.bib" data-xmlurl="/xml/01087063.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/01087063.pdf" title="Zbl 0883.65088 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/S0045-7825(96)01132-2" aria-label="DOI for “Moving least-square reproducing kernel methods. I: Methodology and convergence”" title="10.1016/S0045-7825(96)01132-2">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">Gingold, R. A.; Monaghan, J. J., Kernel estimates as a basis for general particle methods in hydrodynamics, J. Comput. Phys., 46, 429-453 (1982) · <a href="/0487.76010" class="nowrap">Zbl 0487.76010</a></td> </tr><tr> <td>[2]</td> <td class="space">Monaghan, J. J., Why particle methods work, SIAM J. Scient. Stat. Comput., 3, 422-433 (1982) · <a href="/0498.76010" class="nowrap">Zbl 0498.76010</a></td> </tr><tr> <td>[3]</td> <td class="space">Nayroles, B.; Touzot, G.; Villon, P., Generalizing the finite element method: Diffuse approximation and diffuse elements, Comput. Mech., 10, 307-318 (1992) · <a href="/0764.65068" class="nowrap">Zbl 0764.65068</a></td> </tr><tr> <td>[4]</td> <td class="space">Belytschko, T.; Lu, Y. Y.; Gu, L., Element free Galerkin methods, Int. J. Numer. Methods Engrg., 37, 229-256 (1994) · <a href="/0796.73077" class="nowrap">Zbl 0796.73077</a></td> </tr><tr> <td>[5]</td> <td class="space">Belytschko, T.; Gu, L.; Lu, Y. Y., Fracture and crack growth by element free Galerkin methods, Modelling Simul. Mater. Sci. Engrg., 2, 519-534 (1994)</td> </tr><tr> <td>[6]</td> <td class="space">Liu, W. K.; Jun, S.; Zhang, Y., Reproducing kernel particle methods, Int. J. Numer. Methods Fluids, 20, 1081-1106 (1995) · <a href="/0881.76072" class="nowrap">Zbl 0881.76072</a></td> </tr><tr> <td>[7]</td> <td class="space">Liu, W. K.; Chen, Y., Wavelet and multiple scale reproducing kernel methods, Int. J. Numer. Methods Fluids, 21, 901-933 (1995) · <a href="/0885.76078" class="nowrap">Zbl 0885.76078</a></td> </tr><tr> <td>[8]</td> <td class="space">Liu, W. K.; Jun, S.; Li, S.; Adee, J.; Belytschko, T., Reproducing kernel particle methods for structural dynamics, Int. J. Numer. Methods Engrg., 38, 1655-1679 (1995) · <a href="/0840.73078" class="nowrap">Zbl 0840.73078</a></td> </tr><tr> <td>[9]</td> <td class="space">Liu, W. K., An introduction to wavelet reproducing kernel particle methods, USACM Bulletin, 8, 1, 3-16 (1995)</td> </tr><tr> <td>[10]</td> <td class="space">Lancaster, P.; Salkauskas, K., Surfaces generated by moving least square methods, Math. Comput., 37, 155, 141-158 (1980) · <a href="/0469.41005" class="nowrap">Zbl 0469.41005</a></td> </tr><tr> <td>[11]</td> <td class="space">Li, S.; Liu, W. K., Moving least square reproducing kernel method. Part II: Fourier analysis, Comput. Methods Appl. Mech. Engrg., 139, 159-194 (1996) · <a href="/0883.65089" class="nowrap">Zbl 0883.65089</a></td> </tr><tr> <td>[12]</td> <td class="space">Wells, R. O.; Zhou, X., Wavelet interpolation and approximate solutions of elliptic partial differential equations (1993), preprint</td> </tr><tr> <td>[13]</td> <td class="space">Duarte, C. A.; Oden, J. T., <span class="zbmathjax-textit">Hp</span> Clouds—a meshless method to solve boundary-value problems (1995), preprint</td> </tr><tr> <td>[14]</td> <td class="space">Rudin, W., Principles of Mathematical Analysis (1976), McGraw-Hill, Inc · <a href="/0148.02903" class="nowrap">Zbl 0148.02903</a></td> </tr><tr> <td>[15]</td> <td class="space">Mitrinovic̀, D. S., Analytic Inequalities (1970), Springer-Verlag: Springer-Verlag Berlin, Heidelberg, New York · <a href="/0199.38101" class="nowrap">Zbl 0199.38101</a></td> </tr><tr> <td>[16]</td> <td class="space">Davis, P. J., Interpolation and Approximation (1975), Dover Publications, Inc: Dover Publications, Inc New York · <a href="/0111.06003" class="nowrap">Zbl 0111.06003</a></td> </tr><tr> <td>[17]</td> <td class="space">Chui, C. K., An Introduction to Wavelet (1992), Academic Press: Academic Press New York · <a href="/0925.42016" class="nowrap">Zbl 0925.42016</a></td> </tr><tr> <td>[18]</td> <td class="space">Babuška, I.; Melenk, J. M., The partition of unity finite element method, (Technical Note BN-1185 (1995), Institute for Physical Science and Technology, University of Maryland) · <a href="/0949.65117" class="nowrap">Zbl 0949.65117</a></td> </tr><tr> <td>[19]</td> <td class="space">de Boor, C.; Fix, G. J., Spline approximation by quasi-interpolants, J. Approx. Theory, 8, 19-45 (1975) · <a href="/0279.41008" class="nowrap">Zbl 0279.41008</a></td> </tr><tr> <td>[20]</td> <td class="space">Horn, R. A.; Johnson, C. R., Matrix Analysis (1985), Cambridge University Press · <a href="/0576.15001" class="nowrap">Zbl 0576.15001</a></td> </tr><tr> <td>[21]</td> <td class="space">Golub, G. H.; Van Loan, C. F., Matrix Computations (1989), The Johns Hopkins University Press: The Johns Hopkins University Press Baltimore, MD · <a href="/0733.65016" class="nowrap">Zbl 0733.65016</a></td> </tr><tr> <td>[22]</td> <td class="space">Ciarlet, P. G., The Finite Element Method For Elliptic Problems (1978), North-Holland Publishing Company: North-Holland Publishing Company Amsterdam, New York · <a href="/0445.73043" class="nowrap">Zbl 0445.73043</a></td> </tr><tr> <td>[23]</td> <td class="space">Babuška, I., The finite element method with Lagrangian multipliers, Numer. Math., 20, 179-192 (1973) · <a href="/0258.65108" class="nowrap">Zbl 0258.65108</a></td> </tr><tr> <td>[24]</td> <td class="space">Hughes, T. J.R., The Finite Element Method: Linear Static and Dynamic Finite Element Analysis (1987), Prentice Hall: Prentice Hall Englewood Cliffs, NJ · <a href="/0634.73056" class="nowrap">Zbl 0634.73056</a></td> </tr><tr> <td>[25]</td> <td class="space">Rachford, H. H.; Wheeler, M. F., An $H^{−1}$-Galerkin procedure for the two-point boundary value, (de Boor, C., Mathematical Aspects of Finite Elements in Partial Differential Equations (1974), Academic Press: Academic Press New York) · <a href="/0347.65037" class="nowrap">Zbl 0347.65037</a></td> </tr><tr> <td>[26]</td> <td class="space">Babuška, I.; Oden, J. T.; Lee, J. K., Mixed-hybrid finite element approximations of second-order elliptic boundary-value problems, Comput. Methods Appl. Mech. Engrg., 11, 175-206 (1977) · <a href="/0382.65056" class="nowrap">Zbl 0382.65056</a></td> </tr><tr> <td>[27]</td> <td class="space">Daubechies, I., Wavelets, CBMS-NSF Series in Applied Mathematics (1992), SIAM Publication: SIAM Publication Philadelphia · <a href="/0776.42018" class="nowrap">Zbl 0776.42018</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. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching. </div> </div></article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; </script> <script src="/static/scripts.js?v=20240926"> </script> <script src="https://static.zbmath.org/contrib/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ "HTML-CSS": { preferredFont: "TeX", availableFonts: [ "STIX", "TeX" ], linebreaks: { automatic: true }, EqnChunk: (MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { Macros: { Aut: "\\operatorname{Aut}", Hom: "\\operatorname{Hom}" }, noUndefined: { attributes: { mathcolor: "#039", //"red", mathbackground: "white", //"#FFEEEE", mathsize: "90%" } } }, messageStyle: "none" }); </script> <script type="text/javascript"> $(document).ready(function() { $("#MathInput").stop(true, true).keyup(function() { $.ajax({ url: "/mwsq/", type: "POST", data: { query : $("#MathInput").val() }, dataType: "text" }) .done(function(xml) { $("#MathPreview").html(xml); $(window).resize(); }); }); var press = jQuery.Event("keyup"); press.ctrlKey = false; press.which = 40; $("#MathInput").trigger(press); }); </script> <div id="new_tab_icon" style="display: none"> <span class="glyphicon glyphicon-new-window" aria-hidden="true"></span><span class="sr-only">(opens in new tab)</span></div> </body> </html>

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