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E-book: 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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  • Series: Lecture Notes in Mathematics 1939
  • Pub. Date: 01-Apr-2008
  • Publisher: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Language: eng
  • ISBN-13: 9783540783190
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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, 2006Larger Image
  • Format: PDF+DRM
  • Series: Lecture Notes in Mathematics 1939
  • Pub. Date: 01-Apr-2008
  • Publisher: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Language: eng
  • ISBN-13: 9783540783190
Other books in subject:

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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 v
Mixed Finite Element Methods
1(44)
Ricardo G. Duran
Introduction
1(1)
Preliminary Results
2(6)
Mixed Approximation of Second Order Elliptic Problems
8(17)
A Posteriori Error Estimates
25(9)
The General Abstract Setting
34(11)
References
42(3)
Finite Elements for the Stokes Problem
45(56)
Daniele Boffi
Franco Brezzi
Michel Fortin
Introduction
45(1)
The Stokes Problem as a Mixed Problem
46(4)
Mixed Formulation
46(4)
Some Basic Examples
50(6)
Standard Techniques for Checking the Inf-Sup Condition
56(10)
Fortin's Trick
56(1)
Projection onto Constants
57(1)
Verfurth's Trick
58(2)
Space and Domain Decomposition Techniques
60(1)
Macroelement Technique
61(2)
Making Use of the Internal Degrees of Freedom
63(3)
Spurious Pressure Modes
66(3)
Two-Dimensional Stable Elemnts
69(4)
The MINI Element
69(1)
The Crouzeix-Raviart Elemnt
70(1)
P1NC-P0 Approximation
71(1)
Qk-Pk-1 Elements
72(1)
Three-Dimensional Elements
73(2)
The MINI Element
73(1)
The Crouseix-Raviart Element
74(1)
P1NC-P0 Approximation
74(1)
Qk-Pk-1 Elements
75(1)
Pk-Pk-1 Schemes and Generalized Hood-Taylor Elements
75(10)
Pk-Pk-1 Elements
75(1)
Generalized Hood-Taylor Elements
76(9)
Nearly Incompressible Elasticity, Reduced Intergraion Methods and Relation with Penalty Methods
85(7)
Variational Formulations and Admissible Discretizations
85(1)
Reduced Integration Methods
86(2)
Effects of Inexact Integration
88(4)
Divergence-Free Basis, Discrete Stream Functions
92(4)
Other Mixed and Hybrid Methods for Incompressible Flows
96(5)
References
97(4)
Polynomial Exact Sequences and Projection-Based Interpolation with Application to Maxwell Equations
101(94)
Leszek Demkowicz
Introduction
101(1)
Exact Polynomial Sequences
102(13)
One-Dimensional Sequences
102(3)
Two-Dimensional Sequences
105(10)
Commuting Projections and Projection-Based Interpolation Operators in One Space Dimension
115(13)
Commuting Projections: Projection Error Estimates
115(2)
Commuting Interpolation Operators: Interpolation Error Estimates
117(8)
Localization Argument
125(3)
Commuting Projections and Projection-Based Interpolation Operators in Two Space Dimensions
128(13)
Definitions and Commutativity
128(3)
Polynomial Preserving Extension Operators
131(1)
Rught-Inverse of the Curl Operator: Discrete Friedrichs Inequality
132(3)
Projection Error Estimates
135(2)
Interpolation Error Estimates
137(2)
Localization Argument
139(2)
Commuting Projections and Projection-Based Interpolation Operators in Three Space Dimenisons
141(11)
Definitions and Commutativity
141(4)
Polynomial Preserving Extension Oprators
145(1)
Polynomail Preserving Right-BAsed Inverses of Grad, Curl, and Div Operators: discrete Friedrichs Inequalities
145(4)
Projection and Interpolation Error Estimates
149(3)
Application to Maxwell Equations: Open Problems
152(7)
Time-Harmonic MAxwell Equatons
152(3)
So Why Does the Projection-Based Interpolation Matter?
155(1)
Open Problems
155(1)
References
156(3)
Finite Element Methods for Linear Elasticity
Richard S. Falk
Introductiion
159(3)
Finite Element Methods with Strong Symmetry
162(5)
Composite Elements
162(2)
Noncomposite Elements of Arnold and Winther
164(3)
Exterior Calculus on Rn
167(3)
Differential Forms
167(3)
Basic Finite Element Spaces and Their Properties
170(7)
Differential Froms with Values in a Vector Space
173(4)
Mixed formulation of the Equation of Elasticity with Weak Symmetry
177(2)
From the de Rham Complex to an Elasticity Complex with Weak Wymmetry
179(1)
Well-Posedness of the Weak Symmetry Formulation of Elasticity
180(2)
Conditions for Stable Appproximation Schemes
182(2)
Stability of Finite Element Approximation Schemes
184(1)
Refined Error Estimates
185(2)
Example of Stable Finite Element Methods for the Weak Symmetry Formulation of Elasticity
187(8)
Arnold, Falk, WintherFamilies
187(1)
Arnold, Falk, Wnther Reduced Elements
188(2)
PEERS
190(1)
A PEERS-Like Method with Improved Stress Approximation
191(1)
Methods of Stenberg
191(1)
Methods of Stenberg
191(4)
References
Finite Elements for the Reissner-Mindlin Plate
195(38)
Richadrd S. Falk
Introduction
195(1)
A Variational Approach to Dimensional Reduction
196(3)
The First Variationl Approach
196(2)
An Alternative Variational Aproach
198(1)
The Reissneer-Mindlin Model
199(1)
Properties of the Solution
200(1)
Regularity Results
201(2)
Finite Element Discretizations
203(1)
Abstract Error Estimates
204(3)
Applications of the Abstract Error Estimates
207(14)
The Duran-Liberman Element(33)
208(2)
The MITC Trinagular Families
210(3)
The Flak-Tu Elements Wtih Discontinous Shear Stresses[ 35]
213(3)
Linked Interpolation Methods
216(2)
The Nonconforming Elemtnt Of Arnold and Falk [ 11]
218(3)
Some Rectangular Reissner-Mindlin Elements
221(3)
Rectangular MITC Elements and Generalizations [ 20, 17, 23, 48]
221(2)
DL4 Method[ 31]
223(1)
Ye's Method
223(1)
Extension to Quadrilaterals
224(1)
Other Approaches
225(5)
Expanded Mixed Formulations
225(1)
Simple Modification of the Reissner-Mindlin Energy
225(1)
Least-Squares Stabilizatiion Schemes
226(1)
Discontinuous Galerkin Methods [ 9], [ 8]
227(2)
Methods Using Nonconforming Finite Elements
229(1)
A Negative-Norm Least Squares Method
230(1)
Summary
230(3)
References
230(3)
List of Participants
233
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