Trivariate Local Lagrange Interpolation and Macro Elements of Arbitrary Smoothness PDF
Preview Trivariate Local Lagrange Interpolation and Macro Elements of Arbitrary Smoothness
Trivariate Local Lagrange Interpolation and Macro Elements of Arbitrary Smoothness Michael A. Matt Trivariate Local Lagrange Interpolation and Macro Elements of Arbitrary Smoothness Foreword by Prof. Dr. Ming-Jun Lai RESEARCH Michael A. Matt Mannheim, Germany Dissertation Universität Mannheim, 2011 ISBN 978-3-8348-2383-0 ISBN 978-3-8348-2384-7 (eBook) DOI 10.1007/978-3-8348-2384-7 The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.d-nb.de. Springer Spektrum © Vieweg+Teubner Verlag | Springer Fachmedien Wiesbaden 2012 This work is subject to copyright. 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Springer DE is part of Springer Science+Business Media. www.springer-spektrum.de Foreword Multivariate splines are multivariate piecewise polynomial functions de- finedoveratriangulation(inR2)ortetrahedralpartition(inR3)orasim- plicial partition (in Rd,d≥3) with a certain smoothness. These functions areveryflexibleandcanbeusedtoapproximateanyknownorunknown complicatedfunctions. Usuallyonecandivideanydomainofinterestinto triangle/tetrahedron/simplex pieces according to a given function to be approximated and then using piecewise polynomials to approximate the givenfunction. Theyarealsocomputercompatibleinthesenseofcompu- tation, asonlyadditionsandmultiplicationsareneededtoevaluatethese functionsonacomputer. Thus,theyareamostefficientandeffectivetool fornumericallyapproximatingknownorunknownfunctions. Inaddition, multivariatesplinefunctionsareveryimportanttoolsintheareaofapplied mathematics as they are widely used in approximation theory, computer aidedgeometricdesign,scattereddatafitting/interpolation,numericalso- lutionofpartialdifferentialequationssuchasflowsimulationandimage processingincludingimagedenoising. These functions have been studied in the last fifty years. In particular, univariate splines have been studied thoroughly. One understands uni- variate splines very well in theory and computation. One typical mono- graph on univariate spline functions is the well-known book written by Carl de Boor. It is called ”A Practical Guide to Splines” published first in 1978 by Springer Verlag. Another popular book is Larry Schumaker’s monograph”SplineFunctions: BasicsandApplications”whichisnowin second edition published by Cambridge University Press in 2008. The- ory and computation of bivariate splines are also understood very well. For approximation properties of bivariate splines, there is a monograph "Spline Functions on Triangulation" authored by Ming-Jun Lai and Larry Schumaker, published by Cambridge University Press in 2007. For us- ing bivariate splines for scattered data fitting/interpolation and numeri- cal solution of partial differential equations, one can find the paper by G. Awanou,Ming-JunLaiandP.Wenstonin2006useful. Intheirpaper,they explainedhowtocomputebivariatesplineswithoutconstructingexplicit basis functions and use them for data fitting and numerical solutions of VI Foreword PDEs including 2D Navier-Stokes equations and many flow simulations andimagedenoisingsimulations. Thesenumericalresultscanbefoundat www.math.uga.edu/˜mjlai. Approximation properties of spline functions defined on spherical tri- angulations can also be found in the monograph by M.-J. Lai and L. L. Schumaker mentioned above. A numerical implementation of spherical splinefunctionstoreconstructgeopotentialaroundtheEarthcanbefound in the paper by V. Baramidze, M.-J. Lai, C. K. Shum, and P. Wenston in 2009. In addition, some basic properties of trivariate spline functions can alsobefoundinthemonographbyLaiandSchumaker. Forexample,Lai and Schumaker describe in their monograph how to construct C1 and C2 macro-elementfunctionsovertheAlfeldsplitoftetrahedralpartitionsand theWorsey-Farinsplitoftetrahedralpartitions.However,howtoconstruct locallysupportedbasisfunctionssuchasmacro-elementsforsmoothness > r 2wasnotdiscussedinthemonograph. Inaddition,howtofindinter- polatingsplinefunctionsusingtrivariatesplinefunctionsoflowerdegree withoutusinganyvariationalformulationisnotknown. In his thesis, these two questions are carefully examined and fully an- swered. That is, Michael Matt considers trivariate macro-elements and trivariate local Lagrange interpolation methods. The macro-elements of arbitrary smoothness, based on the Alfeld and the Worsey-Farin split of tetrahedra,aredescribedverycarefullyandareillustratedwithmanyex- amples. MichaelMatthasspentagreatdealoftimeandpatiencetowrite down in detail how to determine for Cr macro-elements for r = 3,4,5,6 and etc.. One has to point out that macro-elements are locally supported functions and they form a superspline subspace which has much smaller dimension than the whole spline space defined on the same tetrahedral partition. As this superspline subspace possesses the same full approxi- mationpowerasthewholesplinespace, theyareextremelyimportantas these are enough to use for approximate known and unknown functions with a much smaller dimension. Thus, they are most efficient. The re- sultsinthisthesisareanimportantextensionoftheliteratureontrivariate macro-elements known so far. Michael Matt also describes two methods forlocalLagrangeinterpolationwithtrivariatesplines. Bothmethods,for oncecontinuouslydifferentiablecubicsplinesbasedonatype-4partition andforsplinesofdegreenineonarbitrarytetrahedralpartitions,areexam- ined very detailed. Especially the method based on arbitrary tetrahedral partitionsisacrucialcontributiontothisareaofresearchsinceitisthefirst Foreword VII method for trivariate Lagrange interpolation for two times continuously differentiablesplines. To summarize, the results in this dissertation extend the construction theory of trivariate splines of Cr macro-elements for any r≥1. The con- structionschemesforfindinginterpolatorysplinesoverarbitrarytetrahe- dralpartitionpresentedinthisthesisarehighlyrecommendedforanyap- plicationpractitioner. Prof. Dr. Ming-JunLai DepartmentofMathematics UniversityofGeorgia Athens,GA,U.S.A. Acknowledgments IwouldliketothankProf. Dr. GüntherNürnbergerandProf. Dr. Ming- JunLaifortheirsupportinwritingthisdissertation. Iwanttothankthem for interesting discussions and suggestions that helped to improve this work. Ialsowouldliketothankmyparents, mysister, andallotherper- sons who encouraged and supported me while writing this dissertation andaccompaniedmeinthistime. MichaelAndreasMatt Abstract In this work, we construct two trivariate local Lagrange interpolation methodswhichyieldoptimalapproximationorderandCrmacro-elements based on the Alfeld and the Worsey-Farin split of a tetrahedral partition. The first interpolation method is based on cubic C1 splines over type-4 cube partitions, for which numerical tests are given. The other one is the firsttrivariateLagrangeinterpolationmethodusingC2 splines. Itisbased onarbitrarytetrahedralpartitionsusingsplinesofdegreenine. Inorderto obtainthismethod,severalnewresultsonC2 splinesoverpartialWorsey- Farinsplitsarerequired. Weconstructtrivariatemacro-elementsbasedon theAlfeld,whereeachtetrahedronisdividedintofoursubtetrahedra,and theWorsey-Farinsplit,whereeachtetrahedronisdividedintotwelvesub- tetrahedra,ofatetrahedralpartition.Inordertoobtainthemacro-elements basedontheWorsey-Farinsplitweconstructminimaldeterminingsetsfor Cr macro-elements over the Clough-Tocher split of a triangle, which are morevariablethanthoseintheliterature. Zusammenfassung In dieser Arbeit konstruieren wir zwei Methoden zur lokalen Lagrange- InterpolationmittrivariatenSplines, welcheoptimaleApproximationsor- dnungbesitzen,sowieCrMakro-Elemente,welcheaufderAlfeld-undder Worsey-Farin-Unterteilung einer Tetraederpartition basieren. Eine Inter- polationsmethodebasiertaufkubischenC1SplinesaufTyp-4Würfelparti- tionen. Für diese werden auch numerische Tests angegeben. Die nächste Methode ist die erste zur lokalen Lagrange-Interpolation mit trivariaten C2 Splines. SiebasiertaufbeliebigenTetraederpartitionenundverwendet Splines vom Grad neun. Um diese Methode zu erhalten, werden einige neueResultatezuC2SplinesüberpartiellenWorsey-Farin-Unterteilungen benötigt. WirkonstruierentrivariateMakro-ElementebeliebigerGlattheit, dieaufderAlfeld-Unterteilung,beiderjederTetraederinvierSubtetraeder unterteiltist,undderWorsey-Farin-Unterteilung,beiwelcherjederTetra- ederinzwölfSubtetraederunterteiltist,einerTetraederpartitionberuhen. UmdieMakro-Elemente,dieaufderWorsey-Farin-Unterteilungbasieren, zu erzeugen, konstruieren wir minimal bestimmende Mengen für Cr Makro-Elemente über der Clough-Tocher-Unterteilung eines Dreiecks, welche, im Vergleich zu den bereits in der Literatur bekannten, variabler sind.
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