Ebook: Conjugate Gradient Algorithms and Finite Element Methods

The position taken in this collection of pedagogically written essays is that conjugate gradient algorithms and finite element methods complement each other extremely well. Via their combinations practitioners have been able to solve differential equations and multidimensional problems modeled by ordinary or partial differential equations and inequalities, not necessarily linear, optimal control and optimal design being part of these problems. The aim of this book is to present both methods in the context of complicated problems modeled by linear and nonlinear partial differential equations, to provide an in-depth discussion on their implementation aspects. The authors show that conjugate gradient methods and finite element methods apply to the solution of real-life problems. They address graduate students as well as experts in scientific computing.

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Content:
Front Matter....Pages I-XV
Front Matter....Pages 1-1
The Founders of the Conjugate Gradient Method....Pages 3-10
Conjugate Gradients and Finite Elements — a Golden Jubilee....Pages 11-24
Geometric Interpretations of Conjugate Gradient and Related Methods....Pages 25-43
Front Matter....Pages 45-45
The Convergence of Krylov Methods and Ritz Values....Pages 47-68
An Application of the Shermann-Morrison Formula to the GMRES Method....Pages 69-92
A Parallel CG Solver Based on Domain Decomposition and Non-Smooth Aggregation....Pages 93-102
Deflation in Preconditioned Conjugate Gradient Methods for Finite Element Problems....Pages 103-129
Nonsmooth Equation Method for Nonlinear Nonconvex Optimization....Pages 131-145
Front Matter....Pages 147-147
Nonstandard Nonobtuse Refinements of Planar Triangulations....Pages 149-160
Acute Versus Nonobtuse Tetrahedralizations....Pages 161-170
A Posteriori Error Estimation of “Quantities of Interest” on “Quantity-Adapted” Meshes....Pages 171-181
Front Matter....Pages 183-183
Inversion of Block-Tridiagonal Matrices and Nonnegativity Preservation in the Numerical Solution of Linear Parabolic PDEs....Pages 185-195
On the Nonnegativity Conservation in Semidiscrete Parabolic Problems....Pages 197-210
Iterative Solution Methods of the Maxwell Equations Using Staggered Grid Spatial Discretization....Pages 211-221
Iterative Solution of Linear Variational Problems in Hilbert Spaces: Some Conjugate Gradients Success Stories....Pages 223-245
On the Applicationn of Preconditioning Operators for Nonlinear Elliptic Problems....Pages 247-261
On a Conjugate Gradient/Newton/Penalty Method for the Solution of Obstacle Problems. Application to the Solution of an Eikonal System with Dirichlet Boundary Conditions....Pages 263-283
Front Matter....Pages 285-285
The Use of Bilinear Rectangular Elements in Reconstruction of Panoramic Images....Pages 287-296
Finite Element Discretization and Iterative Solution Techniques for Multiphase Flows in Gas-Liquid Reactors....Pages 297-324
Implicit Flux-Corrected Transport Algorithm for Finite Element Simulation of the Compressible Euler Equations....Pages 325-354
Back Matter....Pages 369-384
Application of the PCG Method in Solution of a Nuclear Reactor Criticality Problem....Pages 355-367

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Content:
Front Matter....Pages I-XV
Front Matter....Pages 1-1
The Founders of the Conjugate Gradient Method....Pages 3-10
Conjugate Gradients and Finite Elements — a Golden Jubilee....Pages 11-24
Geometric Interpretations of Conjugate Gradient and Related Methods....Pages 25-43
Front Matter....Pages 45-45
The Convergence of Krylov Methods and Ritz Values....Pages 47-68
An Application of the Shermann-Morrison Formula to the GMRES Method....Pages 69-92
A Parallel CG Solver Based on Domain Decomposition and Non-Smooth Aggregation....Pages 93-102
Deflation in Preconditioned Conjugate Gradient Methods for Finite Element Problems....Pages 103-129
Nonsmooth Equation Method for Nonlinear Nonconvex Optimization....Pages 131-145
Front Matter....Pages 147-147
Nonstandard Nonobtuse Refinements of Planar Triangulations....Pages 149-160
Acute Versus Nonobtuse Tetrahedralizations....Pages 161-170
A Posteriori Error Estimation of “Quantities of Interest” on “Quantity-Adapted” Meshes....Pages 171-181
Front Matter....Pages 183-183
Inversion of Block-Tridiagonal Matrices and Nonnegativity Preservation in the Numerical Solution of Linear Parabolic PDEs....Pages 185-195
On the Nonnegativity Conservation in Semidiscrete Parabolic Problems....Pages 197-210
Iterative Solution Methods of the Maxwell Equations Using Staggered Grid Spatial Discretization....Pages 211-221
Iterative Solution of Linear Variational Problems in Hilbert Spaces: Some Conjugate Gradients Success Stories....Pages 223-245
On the Applicationn of Preconditioning Operators for Nonlinear Elliptic Problems....Pages 247-261
On a Conjugate Gradient/Newton/Penalty Method for the Solution of Obstacle Problems. Application to the Solution of an Eikonal System with Dirichlet Boundary Conditions....Pages 263-283
Front Matter....Pages 285-285
The Use of Bilinear Rectangular Elements in Reconstruction of Panoramic Images....Pages 287-296
Finite Element Discretization and Iterative Solution Techniques for Multiphase Flows in Gas-Liquid Reactors....Pages 297-324
Implicit Flux-Corrected Transport Algorithm for Finite Element Simulation of the Compressible Euler Equations....Pages 325-354
Back Matter....Pages 369-384
Application of the PCG Method in Solution of a Nuclear Reactor Criticality Problem....Pages 355-367
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