Includes supplementary material: [...]

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 eror 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 application 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.