Finite element methods and their applications

Cover

Finite Element Methods and Their Applications

ISBN-10 3540240780 e-ISBN-13 9783540240785

Preface

Contents

1 Elementary Finite Elements

  1.1 Introduction
      1.1.1 A One-Dimensional Model Problem
      1.1.2 A Two-Dimensional Model Problem
      1.1.3 An Extension to General Boundary Conditions
      1.1.4 Programming Considerations
  1.2 Sobolev Spaces
      1.2.1 Lebesgue Spaces
      1.2.2 Weak Derivatives
      1.2.3 Sobolev Spaces
      1.2.4 Poincare\'s Inequality
      1.2.5 Duality and Negative Norms
  1.3 Abstract Variational Formulation
      1.3.1 An Abstract Formulation
      1.3.2 The Finite Element Method
      1.3.3 Examples
  1.4 Finite Element Spaces
      1.4.1 Triangles
      1.4.2 Rectangles
      1.4.3 Three Dimensions
      1.4.4 A C1 Element
  1.5 General Domains
  1.6 Quadrature Rules
  1.7 Finite Elements for Transient Problems
      1.7.1 A One-Dimensional Model Problem
      1.7.2 A Semi-Discrete Scheme in Space
      1.7.3 Fully Discrete Schemes
  1.8 Finite Elements for Nonlinear Problems
      1.8.1 Linearization Approaches
      1.8.2 Implicit Time Approximations
      1.8.3 Explicit Time Approximations
  1.9 Approximation Theory
      1.9.1 Interpolation Errors
      1.9.2 Error Estimates for Elliptic Problems
      1.9.3 L2-Error Estimates
  1.10 Linear System Solution Techniques
      1.10.1 Gaussian Elimination
      1.10.2 The Conjugate Gradient Algorithm
  1.11 Bibliographical Remarks
  1.12 Exercises

2 Nonconforming Finite Elements

  2.1 Second-Order Problems
      2.1.1 Nonconforming Finite Elements on Triangles
      2.1.2 Nonconforming Finite Elements on Rectangles
      2.1.3 Nonconforming Finite Elements on Tetrahedra
      2.1.4 Nonconforming Finite Elements on Parallelepipeds
      2.1.5 Nonconforming Finite Elements on Prisms
  2.2 Fourth-Order Problems
      2.2.1 The Morley Element
      2.2.3 The Zienkiewicz Element
      2.2.4 The Adini Element
  2.3 Nonlinear Problems
  2.4 Theoretical Considerations
      2.4.1 An Abstract Formulation
      2.4.2 Applications
  2.5 Bibliographical Remarks
  2.6 Exercises

3 Mixed Finite Elements

  3.1 A One-Dimensional Model Problem
  3.2 A Two-Dimensional Model Problem
  3.3 Extension to Boundary Conditions of Other Types
      3.3.1 A Neumann Boundary Condition
      3.3.2 A Boundary Condition of Third Type
  3.4 Mixed Finite Element Spaces
      3.4.1 Mixed Finite Element Spaces on Triangles
      3.4.2 Mixed Finite Element Spaces on Rectangles
      3.4.3 Mixed Finite Element Spaces on Tetrahedra
      3.4.4 Mixed Finite Element Spaces on Parallelepipeds
      3.4.5 Mixed Finite Element Spaces on Prisms
  3.5 Approximation Properties
  3.6 Mixed Methods for Nonlinear Problems
  3.7 Linear System Solution Techniques
      3.7.1 Introduction
      3.7.2 The Uzawa Algorithm
      3.7.3 The Minimum Residual Iterative Algorithm
      3.7.4 Alternating Direction Iterative Algorithms
      3.7.5 Mixed-Hybrid Algorithms
      3.7.6 An Equivalence Relationship
  3.8 Theoretical Considerations
      3.8.1 An Abstract Formulation
      3.8.2 The Mixed Finite Element Method
      3.8.3 Examples
      3.8.4 Construction of Projection Operators
      3.8.5 Error Estimates
  3.9 Bibliographical Remarks
  3.10 Exercises

4 Discontinuous Finite Elements

  4.1 Advection Problems
      4.1.1 DG Methods
      4.1.2 Stabilized DG Methods
  4.2 Diffusion Problems
      4.2.1 Symmetric DG Method
      4.2.2 Symmetric Interior Penalty DG Method
      4.2.3 Non-Symmetric DG Method
      4.2.4 Non-Symmetric Interior Penalty DG Method
      4.2.5 Remarks
  4.3 Mixed Discontinuous Finite Elements
      4.3.1 A One-Dimensional Problem
      4.3.2 Multi-Dimensional Problems
      4.3.3 Nonlinear Problems
  4.4 Theoretical Considerations
      4.4.1 DG Methods
      4.4.2 Stabilized DG Methods
  4.5 Bibliographical Remarks
  4.6 Exercises

5 Characteristic Finite Elements

  5.1 An Example
  5.2 The Modified Method of Characteristics
      5.2.1 A One-Dimensional Model Problem
      5.2.2 Periodic Boundary Conditions
      5.2.3 Extension to Multi-Dimensional Problems
      5.2.4 Discussion of a Conservation Relation
  5.3 The Eulerian-Lagrangian Localized Adjoint Method
      5.3.1 A One-Dimensional Model Problem
      5.3.2 Extension to Multi-Dimensional Problems
  5.4 The Characteristic Mixed Method
  5.5 The Eulerian-Lagrangian Mixed Discontinuous Method
  5.6 Nonlinear Problems
  5.7 Remarks on Characteristic Finite Elements
  5.8 Theoretical Considerations
  5.9 Bibliographical Remarks
  5.10 Exercises

6 Adaptive Finite Elements

  6.1 Local Grid Refinement in Space
      6.1.1 Regular H-Schemes
      6.1.2 Irregular H-Schemes
      6.1.3 Unrefinements
  6.2 Data Structures
  6.3 A-Posteriori Error Estimates for Stationary Problems
      6.3.1 Residual Estimators
      6.3.2 Local Problem-Based Estimators
      6.3.3 Averaging-Based Estimators
      6.3.4 Hierarchical Basis Estimators
      6.3.5 Efficiency of Error Estimators
  6.4 A-Posteriori Error Estimates for Transient Problems
  6.5 A-Posteriori Error Estimates for Nonlinear Problems
  6.6 Theoretical Considerations
      6.6.1 An Abstract Theory
      6.6.2 Applications
  6.7 Bibliographical Remarks
  6.8 Exercises

7 Solid Mechanics

  7.1 Introduction
      7.1.1 Kinematics
      7.1.2 Equilibrium
      7.1.3 Material Laws
  7.2 Variational Formulations
      7.2.1 The Displacement Form
      7.2.2 The Mixed Form
  7.3 Finite Element Methods
      7.3.1 Finite Elements and Locking E.ects
      7.3.2 Mixed Finite Elements
      7.3.3 Nonconforming Finite Elements
  7.4 Theoretical Considerations
  7.5 Bibliographical Remarks
  7.6 Exercises

8 Fluid Mechanics

  8.1 Introduction
  8.2 Variational Formulations
      8.2.1 The Galerkin Approach
      8.2.2 The Mixed Formulation
  8.3 Finite Element Methods
      8.3.1 Galerkin Finite Elements
      8.3.2 Mixed Finite Elements
      8.3.3 Nonconforming Finite Elements .
  8.4 The Navier-Stokes Equation
  8.5 Theoretical Considerations
  8.6 Bibliographical Remarks
  8.7 Exercises

9 Fluid Flow in Porous Media

  9.1 Two-Phase Immiscible Flow
      9.1.1 The Phase Formulation
      9.1.2 The Weighted Formulation
      9.1.3 The Global Formulation
  9.2 Mixed Finite Elements for Pressure
  9.3 Characteristic Methods for Saturation
  9.4 A Numerical Example
  9.5 Theoretical Considerations
      9.5.1 Analysis for the Pressure Equation
      9.5.2 Analysis for the Saturation Equation
  9.6 Bibliographical Remarks
  9.7 Exercises

10 Semiconductor Modeling

  10.1 Three Semiconductor Models
      10.1.1 The Drift-Diffusion Model
      10.1.2 The Hydrodynamic Model
      10.1.3 The Quantum Hydrodynamic Model
  10.2 Numerical Methods
      10.2.1 The Drift-Diffusion Model
      10.2.2 The Hydrodynamic Model
      10.2.3 The Quantum Hydrodynamic Model
  10.3 A Numerical Example
  10.4 Bibliographical Remarks
  10.5 Exercises

A Nomenclature

References

Index