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Full Description
Research on non-standard finite element methods is evolving rapidly and in this text Brezzi and Fortin give a general framework in which the development is taking place. The presentation is built around a few classic examples: Dirichlet's problem, Stokes problem, Linear elasticity. The authors provide with this publication an analysis of the methods in order to understand their properties as thoroughly as possible.
Contents
I: Variational Formulations and Finite Element Methods.- §1. Classical Methods.- §2. Model Problems and Elementary Properties of Some Functional Spaces.- §3. Duality Methods.- §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.- §2. Approximation of the Problem.- §3. Numerical Properties of the Discrete Problem.- §4. Solution by Penalty Methods, Convergence of Regularized Problems.- §5. Iterative Solution Methods. Uzawa's Algorithm.- §6. Concluding Remarks.- III: Function Spaces and Finite Element Approximations.- §1. Properties of the spaces Hs(?) and H(div; ?).- §2. Finite Element Approximations of H1(?) and H2(?).- §3. Approximations of H (div; ?).- §4. Concluding Remarks.- IV: Various Examples.- §1. Nonstandard Methods for Dirichlet's Problem.- §2. Stokes Problem.- §3. Elasticity Problems.- §4. A Mixed Fourth-Order Problem.- §5. Dual Hybrid Methods for Plate Bending Problems.- V: Complements on Mixed Methods for Elliptic Problems.- §1. Numerical Solutions.- §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.- §3. Examples of Elements for Incompressible Materials.- §4. Standard Techniques of Proof for the inf-sup Condition.- §5. Macroelement Techniques and Spurious Pressure Modes.- §6. An Alternative Technique of Proof and Generalized Taylor-Hood Element.- §7. Nearly Incompressible Elasticity, Reduced Integration Methods and Relation with Penalty Methods.- §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.- References.
