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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.
The Founders of the Conjugate Gradient Method
Conjugate Gradients and Finite Elements — a Golden Jubilee
Geometric Interpretations of Conjugate Gradient and Related Methods
The Convergence of Krylov Methods and Ritz Values
An Application of the Shermann-Morrison Formula to the GMRES Method
A Parallel CG Solver Based on Domain Decomposition and Non-Smooth Aggregation
Deflation in Preconditioned Conjugate Gradient Methods for Finite Element Problems
Nonsmooth Equation Method for Nonlinear Nonconvex Optimization
Nonstandard Nonobtuse Refinements of Planar Triangulations
Acute Versus Nonobtuse Tetrahedralizations
A Posteriori Error Estimation of “Quantities of Interest” on “Quantity-Adapted” Meshes
Inversion of Block-Tridiagonal Matrices and Nonnegativity Preservation in the Numerical Solution of Linear Parabolic PDEs
On the Nonnegativity Conservation in Semidiscrete Parabolic Problems
Iterative Solution Methods of the Maxwell Equations Using Staggered Grid Spatial Discretization
Iterative Solution of Linear Variational Problems in Hilbert Spaces: Some Conjugate Gradients Success Stories
On the Applicationn of Preconditioning Operators for Nonlinear Elliptic Problems
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
The Use of Bilinear Rectangular Elements in Reconstruction of Panoramic Images
Finite Element Discretization and Iterative Solution Techniques for Multiphase Flows in Gas-Liquid Reactors
Implicit Flux-Corrected Transport Algorithm for Finite Element Simulation of the Compressible Euler Equations
Application of the PCG Method in Solution of a Nuclear Reactor Criticality Problem