Mixed Finite Elements, Compatibility Conditions, and Applications: Lectures Given at the C. I. M. E. Summer School Held in Cetraro, Italy, June 26 - July 1, 2006

by Boffi, Daniele; Brezzi, Franco; Demkowicz, Leszek F.; Duran, Ricardo G.; Falk, Richard S.; Fortin, Michel; Boffi, Daniele

ISBN13: 9783540783145

ISBN10: 3540783148

Format: Paperback

Pub. Date: 2008-06-01

Publisher(s): Springer Verlag

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Summary

Since the early 70's, mixed finite elements have been the object of a wide and deep study by the mathematical and engineering communities. The fundamental role of this method for many application fields has been worldwide recognized and its use has been introduced in several commercial codes. An important feature of mixed finite elements is the interplay between theory and application. Discretization spaces for mixed schemes require suitable compatibilities, so that simple minded approximations generally do not work and the design of appropriate stabilizations gives rise to challenging mathematical problems. This volume collects the lecture notes of a C.I.M.E. course held in Summer 2006, when some of the most world recognized experts in the field reviewed the rigorous setting of mixed finite elements and revisited it after more than 30 years of practice. Applications, in this volume, range from traditional ones, like fluid-dynamics or elasticity, to more recent and active fields, like electromagnetism.

Preface	p. V
Mixed Finite Element Methods	p. 1
Introduction	p. 1
Preliminary Results	p. 2
Mixed Approximation of Second Order Elliptic Problems	p. 8
A Posteriori Error Estimates	p. 25
The General Abstract Setting	p. 34
References	p. 42
Finite Elements for the Stokes Problem	p. 45
Introduction	p. 45
The Stokes Problem as a Mixed Problem	p. 46
Mixed Formulation	p. 46
Some Basic Examples	p. 50
Standard Techniques for Checking the Inf-Sup Condition	p. 56
Fortin's Trick	p. 56
Projection onto Constants	p. 57
Verfürth's Trick	p. 58
Space and Domain Decomposition Techniques	p. 60
Macroelement Technique	p. 61
Making Use of the Internal Degrees of Freedom	p. 63
Spurious Pressure Modes	p. 66
Two-Dimensional Stable Elements	p. 69
The MINI Element	p. 69
The Crouzeix-Raviart Element	p. 70
<$>P_1^{NC} - P_0<$> Approximation	p. 71
Q_k − P_k−1 Elements	p. 72
Three-Dirnensional Elements	p. 73
The MINI Element	p. 73
The Crouseix-Raviart Element	p. 74
<$>P_1^{NC} - P_0<$> Approximation	p. 74
Q_k − P_{k − 1} Elements	p. 75
P_k − P_{k − 1} Schemes and Generalized Hood-Taylor Elements	p. 75
P_k − P_{k − 1} Elements	p. 75
Generalized Hood-Taylor Elements	p. 76
Nearly Incompressible Elasticity, Reduced Integration Methods and Relation with Penalty Methods	p. 85
Variational Formulations and Admissible Discretizations	p. 85
Reduced Integration Methods	p. 86
Effects of Inexact Integration	p. 88
Divergence-Free Basis, Discrete Stream Functions	p. 92
Other Mixed and Hybrid Methods for Incompressible Flows	p. 96
References	p. 97
Polynomial Exact Sequences and Projection-Based Interpolation with Application to Maxwell Equations	p. 101
Introduction	p. 101
Exact Polynomial Sequences	p. 102
One-Dimensional Sequences	p. 102
Two-Dimensional Sequences	p. 105
Commuting Projections and Projection-Based Interpolation Operators in One Space Dimension	p. 115
Commuting Projections: Projection Error Estimates	p. 115
Commuting Interpolation Operators: Interpolation Error Estimates	p. 117
Localization Argument	p. 125
Commuting Projections and Projection-Based Interpolation Operators in Two Space Dimensions	p. 128
Definitions and Commutativity	p. 128
Polynomial Preserving Extension Operators	p. 131
Right-Inverse ofthe Curl Operator: Discrete Friedrichs Inequality	p. 132
Projection Error Estimates	p. 135
Interpolation Error Estimates	p. 137
Localization Argument	p. 139
Commuting Projections and Projection-Based Interpolation Operators in Three Space Dimensions	p. 141
Definitions and Commutativity	p. 141
Polynomial Preserving Extension Operators	p. 145
Polynomial Preserving, Right-Inverses of Grad, Curl, and Div Operators: Discrete Friedrichs Inequalities	p. 145
Projection and Interpolation Error Estimates	p. 149
Application to Maxwell Equations: Open Problems	p. 152
Time-Harmonic Maxwell Equations	p. 152
So Why Does the Projection-Based Interpolation Matter?	p. 155
Open Problems	p. 155
References	p. 156
Finite Element Methods for Linear Elasticity	p. 159
Introduction	p. 159
Finite Element Methods with Strong Symmetry	p. 162
CompositeElements	p. 162
Noncomposite Elements of Arnold and Winther	p. 164
Exterior Calculus on <$>{\op R}^n<$>	p. 167
Differentia Forms	p. 167
Basic Finite Element Spaces and their Properties	p. 170
Differential Forms with Values in a Vector Space	p. 173
Mixed Formulation of the Equations of Elasticity with Weak Symmetry	p. 177
From the de Rham Complex to an Elasticity Complex with Weak Symmetry	p. 179
Well-Posedness of the Weak Symmetry Formulation of Elasticity	p. 180
Conditions for Stable Approximation Schemes	p. 182
Stability of Finite Element Approximation Schemes	p. 184
Refined Error Estimates	p. 185
Examples of Stable Finite Element Methods for the Weak Symmetry FormulationofElasticity	p. 187
Arnold, Falk, Winther Families	p. 187
Arnold, Falk, Winther Reduced Elements	p. 188
PEERS	p. 190
A PEERS-Like Method with Improved Stress Approximation	p. 191
Methods of Stenberg	p. 191
References	p. 193
Finite Elements for the Reissner-Mindlin Plate	p. 195
Introduction	p. 195
A Variational Approach to Dimensional Reduction	p. 196
The First Variational Approach	p. 196
An Alternative Variational Approach	p. 198
The Reissner-Mindlin Model	p. 199
Properties of the Solution	p. 200
Regularity Results	p. 201
Finite Element Discretizations	p. 203
Abstract Error Analysis	p. 204
Applications of the Abstract Error Estimates	p. 207
The Durán-Liberman Element [33]	p. 208
The MITC Triangular Families	p. 210
The Falk-Tu Elements With Discontinuous Shear Stresses [35]	p. 213
Linked Interpolation Methods	p. 216
The Nonconforming Element of Arnold and Falk [11]	p. 218
Some Rectangular Reissner-Mindlin Elements	p. 221
Rectangular MITC Elements and Generalizations [20, 17, 23, 48]	p. 221
DL4 Method [31]	p. 223
Ye's Method	p. 223
Extension to Quadrilaterals	p. 224
Other Approaches	p. 225
Expanded Mixed Formulations	p. 225
SimpleModificationoftheReissner-MindlinEnergy	p. 225
Least-Squares Stabilization Schemes	p. 226
Discontinuous Galerkin Methods [9], [8]	p. 227
Methods Using Nonconforming Finite Elements	p. 229
A Negative-Norm Least Squares Method	p. 230
Summary	p. 230
References	p. 230
List of Participants	p. 233
Table of Contents provided by Publisher. All Rights Reserved.

Mixed Finite Elements, Compatibility Conditions, and Applications: Lectures Given at the C. I. M. E. Summer School Held in Cetraro, Italy, June 26 - July 1, 2006

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