Mixed and Hybrid Finite Element Methods

17.07.1991

Englisch

9780387975825

Springer US

Springer Series in Computational Mathematics

Nein

Michel Fortin

350

Gebundene Ausgabe

15

finite element method, calculus, development, Approximation, model

PBKS - Numerische Mathematik

I: Variational Formulations and Finite Element Methods.- §1. Classical Methods.- §2. Model Problems and Elementary Properties of Some Functional Spaces.- §3. Duality Methods.- 3.1. Generalities.- 3.2. Examples for symmetric problems.- 3.3. Duality methods for nonsymmetric bilinear forms.- §4. Domain Decomposition Methods, Hybrid Methods.- §5. Augmented Variational Formulations.- §6. Transposition Methods.- §7. Bibliographical remarks.- II: Approximation of Saddle Point Problems.- §1. Existence and Uniqueness of Solutions.- 1.1. Quadratic problems under linear constraints.- 1.2. Extensions of existence and uniqueness results.- §2. Approximation of the Problem.- 2.1. Basic results.- 2.2. Error estimates for the basic problem.- 2.3. The inf-sup condition: criteria.- 2.4. Extensions of error estimates.- 2.5. Various generalizations of error estimates.- 2.6. Perturbations of the problem, nonconforming methods.- 2.7. Dual error estimates.- §3. Numerical Properties of the Discrete Problem.- 3.1. The matrix form of the discrete problem.- 3.2. Eigenvalue problem associated with the inf-sup condition.- 3.3. Is the inf-sup condition so important?.- §4. Solution by Penalty Methods, Convergence of Regularized Problems.- §5. Iterative Solution Methods. Uzawa’s Algorithm.- 5.1. Standard Uzawa’s algorithm.- 5.2. Augmented Lagrangian algorithm.- §6. Concluding Remarks.- III: Function Spaces and Finite Element Approximations.- §1. Properties of the spaces Hs(?) and H(div; ?).- 1.1. Basic results.- 1.2. Properties relative to a partition of ?.- 1.3. Properties relative to a change of variables.- §2. Finite Element Approximations of H1(?) and H2(?).- 2.1. Conforming methods.- 2.2. Nonconforming methods.- 2.3. Nonpolynomial approximations: Spaces Lks(Eh).- 2.4. Scaling arguments.- §3. Approximations of H (div; ?).- 3.1. Simplicial approximations of H (div; K).- 3.2. Rectangular approximations of H (div; K).- 3.3. Interpolation operator and error estimates.- 3.4. Approximation spaces for H (div; ?).- §4. Concluding Remarks.- IV: Various Examples.- §1. Nonstandard Methods for Dirichlet’s Problem.- 1.1. Description of the problem.- 1.2. Mixed finite element methods for Dirichlet’s problem.- 1.3. Primal hybrid methods.- 1.4. Dual hybrid methods.- §2. Stokes Problem.- §3. Elasticity Problems.- §4. A Mixed Fourth-Order Problem.- 4.1. The ?-? biharmonic problem.- §5. Dual Hybrid Methods for Plate Bending Problems.- V: Complements on Mixed Methods for Elliptic Problems.- §1. Numerical Solutions.- 1.1. Preliminaries.- 1.2. Interelement multipliers.- §2. A Brief Analysis of the Computational Effort.- §3. Error Analysis for the Multiplier.- §4. Error Estimates in Other Norms.- §5. Application to an Equation Arising from Semiconductor Theory.- §6. How Things Can Go Wrong.- §7. Augmented Formulations.- VI: Incompressible Materials and Flow Problems.- §1. Introduction.- §2. The Stokes Problem as a Mixed Problem.- 2.1. Mixed Formulation.- §3. Examples of Elements for Incompressible Materials.- 3.1. Simple examples.- §4. Standard Techniques of Proof for the inf-sup Condition.- 4.1. General results.- 4.2. Higher order methods.- §5. Macroelement Techniques and Spurious Pressure Modes.- 5.1. Some remarks about spurious pressure modes.- 5.2. An abstract convergence result.- 5.3. Macroelement techniques.- 5.4. The bilinear velocity-constant pressure (Q1-P0) element.- 5.5. Other stabilization procedures, (Augmented Formulations).- §6. An Alternative Technique of Proof and Generalized Taylor-Hood Element.- §7. Nearly Incompressible Elasticity, Reduced Integration Methods and Relation with Penalty Methods.- 7.1. Variational formulations and admissible discretizations.- 7.2. Reduced integration methods.- 7.3. Effects of inexact integration.- §8. Divergence-Free Basis, Discrete Stream Functions.- §9. Other Mixed and Hybrid Methods for Incompressible Flows.- VII: Other Applications.- §1. Mixed Methods for Linear Thin Plates.- §2. Mixed Methods for Linear Elasticity Problems.- §3. Moderately Thick Plates.- 3.1. Generalities.- 3.2. Discretization of the problem.- 3.3. Continuous Pressure Approximations.- 3.4. Discontinuous Pressure Approximations.- References.

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