Mixed Finite Elements, Compatibility Conditions, and Applications.pdf

发布时间：2022-06-13 发布人：admin 分类：说明书资料大小：3.16M 资料格式：pdf 举报版权申诉

79e70c69-913f-4343-a86a-5847f9ade127.pdf-第1页.png

第1页 / 共253页

79e70c69-913f-4343-a86a-5847f9ade127.pdf-第2页.png

第2页 / 共253页

79e70c69-913f-4343-a86a-5847f9ade127.pdf-第3页.png

第3页 / 共253页

79e70c69-913f-4343-a86a-5847f9ade127.pdf-第4页.png

第4页 / 共253页

79e70c69-913f-4343-a86a-5847f9ade127.pdf-第5页.png

第5页 / 共253页

79e70c69-913f-4343-a86a-5847f9ade127.pdf-第6页.png

第6页 / 共253页

79e70c69-913f-4343-a86a-5847f9ade127.pdf-第7页.png

第7页 / 共253页

79e70c69-913f-4343-a86a-5847f9ade127.pdf-第8页.png

第8页 / 共253页

front-matter

fulltext

fulltext2

fulltext3

fulltext4

fulltext5

back-matter

Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris

C.I.M.E. means Centro Internazionale Matematico Estivo, that is, International Mathematical Summer Center. Conceived in the early ﬁfties, it was born in 1954 and made welcome by the world mathematical community where it remains in good health and spirit. Many mathematicians from all over the world have been involved in a way or another in C.I.M.E.’s activities during the past years. So they already know what the C.I.M.E. is all about. For the beneﬁt of future potential users and co- operators the main purposes and the functioning of the Centre may be summarized as follows: every year, during the summer, Sessions (three or four as a rule) on different themes from pure and applied mathematics are offered by application to mathematicians from all countries. Each session is generally based on three or four main courses (− hours over a period of - working days) held from specialists of international renown, plus a certain number of seminars. A C.I.M.E. Session, therefore, is neither a Symposium, nor just a School, but maybe a blend of both. The aim is that of bringing to the attention of younger researchers the origins, later developments, and perspectives of some branch of live mathematics. The topics of the courses are generally of international resonance and the participation of the courses cover the expertise of different countries and continents. Such combination, gave an excellent opportu- nity to young participants to be acquainted with the most advance research in the topics of the courses and the possibility of an interchange with the world famous specialists. The full immersion atmosphere of the courses and the daily exchange among participants are a ﬁrst building brick in the ediﬁce of international collaboration in mathematical research. C.I.M.E. Director Pietro ZECCA Dipartimento di Energetica “S. Stecco” Università di Firenze Via S. Marta, 3 50139 Florence Italy e-mail: zecca@uniﬁ.it C.I.M.E. Secretary Elvira MASCOLO Dipartimento di Matematica Università di Firenze viale G.B. Morgagni 67/A 50134 Florence Italy e-mail: mascolo@math.uniﬁ.it For more information see CIME’s homepage: http://www.cime.uniﬁ.it CIME’s activity is supported by: – Istituto Nationale di Alta Mathematica “F. Severi” – Ministero dell’Istruzione, dell’Università e delle Ricerca

Daniele Bofﬁ · Franco Brezzi Leszek F. Demkowicz · Ricardo G. Durán Richard S. Falk · Michel Fortin Mixed Finite Elements, Compatibility Conditions, and Applications Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy June –July , Editors: Daniele Bofﬁ Lucia Gastaldi 123

and Istituto di Matematica Applicata e Tecnologie Informatiche del C.N.R. Via Ferrata 3, 27100 Pavia, Italy brezzi@imati.cnr.it http://www.imati.cnr.it/∼brezzi Leszek F. Demkowicz Institute for Computational Engineering and Sciences Daniele Bofﬁ Dipartimento di Matematica Università degli studi di Pavia Via Ferrata 1, 27100 Pavia, Italy daniele.bofﬁ@unipv.it http://www-dimat.unipv.it/bofﬁ Franco Brezzi Istituto Universitario di Studi Superiori (IUSS) The University of Texas at Austin ACES 6.332, 105 Austin, TX 78712, USA leszek@ices.utexas.edu http://users.ices.utexas.edu/∼leszek Ricardo G. Durán Departamento de Matemática Facultad de Ciencias Exactas y Naturales Universidad de Buenos Aires Ciudad Universitaria. Pabellón I 1428 Buenos Aires, Argentina rduran@dm.uba.ar http://mate.dm.uba.ar/∼rduran Richard S. Falk Department of Mathematics - Hill Center Rutgers, The State University of New Jersey 110 Frelinghuysen Rd. Piscataway, NJ 08854-8019, USA falk@math.rutgers.edu http://www.math.rutgers.edu/∼falk Michel Fortin Département de mathématiques et de statistique Pavillon Alexandre-Vachon Université Laval 1045, avenue de la Médecine Québec (Québec) G1V 0A6, Canada mfortin@giref.ulaval.ca http://www.mat.ulaval.ca Lucia Gastaldi Dipartimento di Matematica Università degli Studi di Brescia Via Valotti 9 25133 Brescia, Italy gastaldi@ing.unibs.it http://dm.ing.unibs.it/gastaldi ISBN: 978-3-540-78314-5 DOI: 10.1007/978-3-540-78319-0 e-ISBN: 978-3-540-78319-0 Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2008921921 Mathematics Subject Classiﬁcation (2000): 65-02, 65N30, 65N12, 35M10, 74S05, 76M10, 78M10, 58A10, 58A12 c 2008 Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September , , in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: WMXDesign GmbH Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com

Preface This volume is a collection of the notes of the C.I.M.E. course “Mixed ﬁnite ele- ments, compatibility conditions, and applications” held in Cetraro (CS), Italy, from June 26 to July 1, 2006. Since the early 1970s, mixed ﬁnite elements have been the object of wide and deep study by the mathematical and engineering communities. The fundamental role of mixed methods for many application ﬁelds has been recognized worldwide and their use has been introduced in several commercial codes. An important feature of mixed ﬁnite elements is the interplay between theory and application: on the one hand, many schemes used for real life simulations have been cast in a rigor- ous framework, and on the other, the theoretical analysis makes it possible to design new schemes or to improve existing ones, based on their mathematical properties. Indeed, due to the compatibility conditions required by the discretization spaces to provide stable schemes, simple minded approximations generally do not work and the design of suitable stabilizations gives rise to challenging mathematical problems. The course had two main goals. The ﬁrst one was to review the rigorous setting of mixed ﬁnite elements and to revisit it after more than 30 years of practice; this resulted in developing a detailed a priori and a posteriori analysis. The second one was to show some examples of possible applications of the method. We are conﬁdent this book will serve as a basic reference for people exploring the ﬁeld of mixed ﬁnite elements. This “Lecture Notes” cover the theory of mixed ﬁnite elements and applications to Stokes problem, elasticity, and electromagnetism. Ricardo G. Dur´an had the responsibility of reviewing the general theory. He started with the description of the mixed approximation of second-order elliptic prob- lems (a priori and a posteriori estimates) and then extended the theory to general mixed problems, thus leading to the famous inf–sup conditions. The second course on Stokes problem has been given by Daniele Bofﬁ, Franco Brezzi, and Michel Fortin. From the basic application of the inf–sup theory to the linear Stokes system, stable Stokes ﬁnite elements have been analyzed, and general stabilization techniques have been described. Finally, some results on visco-elasticity have been presented.

VI Preface Richard S. Falk has dealt with the mixed ﬁnite element approximation of the elasticity problem and, more particularly, of the Reissner–Mindlin plate problem. The corresponding notes are split into two parts: in the ﬁrst one, recent results linking the de Rham complex to ﬁnite element schemes have been reviewed; in the second, classical Reissner–Mindlin plate elements have been presented, together with some discussion on quadrilateral meshes. Leszek Demkowicz has given a general introduction to the exact sequence (de Rham complex) topic, which turns out to be a fundamental tool for the construction and analysis of mixed ﬁnite elements, and for the approximation of problems arising from electromagnetism. The results presented here use special characterization of traces for vector-valued functions in Sobolev spaces. We thank all the lecturers and, in particular, Franco Brezzi, who laid the foun- dation for the analysis of mixed ﬁnite elements, for his active participation in this C.I.M.E. course. Daniele Bofﬁ, Pavia Lucia Gastaldi, Brescia

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V Mixed Finite Element Methods Ricardo G. Dur´an . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Mixed Approximation of Second Order Elliptic Problems . . . . . . . . . . . . . . . 4 A Posteriori Error Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 The General Abstract Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 2 8 25 34 42 Finite Elements for the Stokes Problem Daniele Bofﬁ, Franco Brezzi, and Michel Fortin . . . . . . . . . . . . . . . . . . . . . . . . . 45 45 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Stokes Problem as a Mixed Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 46 2.1 Mixed Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3 Some Basic Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4 Standard Techniques for Checking the Inf–Sup Condition . . . . . . . . . . . . . . . 4.1 Fortin’s Trick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 57 4.2 Projection onto Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Verf¨urth’s Trick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 60 4.4 Space and Domain Decomposition Techniques . . . . . . . . . . . . . . . . . . . . 61 4.5 Macroelement Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Making Use of the Internal Degrees of Freedom . . . . . . . . . . . . . . . . . . . 63 66 5 Spurious Pressure Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 6 Two-Dimensional Stable Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 The MINI Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 6.2 The Crouzeix–Raviart Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 − P0 Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 6.3 P N C 6.4 Qk − Pk−1 Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 1

VIII Contents 7 Three-Dimensional Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 The MINI Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 The Crouseix–Raviart Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . − P0 Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 P N C 7.4 Qk − Pk−1 Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Pk − Pk−1 Schemes and Generalized Hood–Taylor Elements . . . . . . . . . . . . 8.1 Pk − Pk−1 Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Generalized Hood–Taylor Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 9 Nearly Incompressible Elasticity, Reduced Integration Methods and Relation with Penalty Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Variational Formulations and Admissible Discretizations . . . . . . . . . . . 9.2 Reduced Integration Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Effects of Inexact Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Divergence-Free Basis, Discrete Stream Functions . . . . . . . . . . . . . . . . . . . . . 11 Other Mixed and Hybrid Methods for Incompressible Flows . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 73 74 74 75 75 75 76 85 85 86 88 92 96 97 Polynomial Exact Sequences and Projection-Based Interpolation with Application to Maxwell Equations Leszek Demkowicz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 2 Exact Polynomial Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 2.1 One-Dimensional Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 2.2 Two-Dimensional Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 3 Commuting Projections and Projection-Based Interpolation Operators in One Space Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 3.1 Commuting Projections: Projection Error Estimates . . . . . . . . . . . . . . . . 115 3.2 Commuting Interpolation Operators: Interpolation Error Estimates . . . 117 3.3 Localization Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 4 Commuting Projections and Projection-Based Interpolation Operators in Two Space Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 4.1 Deﬁnitions and Commutativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 4.2 Polynomial Preserving Extension Operators . . . . . . . . . . . . . . . . . . . . . . 131 4.3 Right-Inverse of the Curl Operator: Discrete Friedrichs Inequality . . . . 132 4.4 Projection Error Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 4.5 Interpolation Error Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 4.6 Localization Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 5 Commuting Projections and Projection-Based Interpolation Operators in Three Space Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 5.1 Deﬁnitions and Commutativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 5.2 Polynomial Preserving Extension Operators . . . . . . . . . . . . . . . . . . . . . . 145 5.3 Polynomial Preserving, Right-Inverses of Grad, Curl, and Div Operators: Discrete Friedrichs Inequalities . . . . . . . . . . . . . . . . . . . . . . . 145 5.4 Projection and Interpolation Error Estimates . . . . . . . . . . . . . . . . . . . . . . 149

分享到：

赞收藏

资料库

Mixed Finite Elements, Compatibility Conditions, and Applications.pdf

相关推荐

课程资源

热门标签

最新资料