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.
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.
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.
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.
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