Defect Correction Methods: Theory and Applications

The Defect Correction Approach.- Historical Examples of Defect Correction.- General Defect Correction Principles.- Discretization of Operator Equations.- Defect Correction and Discretization.- Multi-level and Multi-grid Methods.- References.- Defect Correction for Operator Equations.- Defect Correction Algorithms for Stiff Ordinary Differential Equations.- Specification of the Algorithm.- Previous Results.- The Concept of B-Convergence.- B-Convergence Properties of Certain IDeC-Algorithms.- References.- On a Principle of Direct Defect Correction Based on A-Posteriori Error Estimates.- Basic Relations Between Defect Corrections and Asymptotic Expansions.- Defect Corrections Through Projection Methods.- Direct Defect Correction via Finite Element Methods for Singularly Perturbed Differential Equations.- References.- Simultaneous Newton's Iteration for the Eigenproblem.- The Sylvester Equation AX?XB= C.- A Quadratic Equation for the Invariant Subspace.- Newton's Method on (7).- Simplified Newton's Method.- Modified Newton's Methods.- Conclusion.- References.- On Some Two-level Iterative Methods.- The General Two-level Iterative Process.- Applications to Integral Equations.- Projection-iterative Methods.- Iterative Aggregation Methods.- Conclusion.- References.- Multi-grid Methods.- Local Defect Correction Method and Domain Decomposition Techniques.- Defect Correction Method.- Local Defect Correction (Algorithm; Numerical Examples of the Local Defect Correction; Error Estimates; Multi-grid Iteration with Local Defect Correction).- Domain Decomposition Methods.- References.- Fast Adaptive Composite Grid (FAC) Methods: Theory for the Variational Case.- Two-level Methods.- Remarks.- References.- Mixed Defect Correction Iteration for the Solution of a Singular Perturbation Problem.- Local Mode Analysis.- The Defect Correction Principle.- The Mixed Defect Correction Process (MDCP).- Local Mode Analysis for the MDCP Solution.- The Convergence of the MDCP Iteration.- Boundary Analysis of the MDCP Solutions.- Numerical Examples.- References.- Computation of Guaranteed High-accuracy Results.- Solution of Linear and Nonlinear Algebraic Problems with Sharp, Guaranteed Bounds.- Computer Arithmetic.- Inclusion Methods for Linear Systems.- Implementation of Inclusion Algorithms.- Nonlinear Systems.- Conclusion.- References.- Residual Correction and Validation in Functoids.- Functoids and Roundings.- Iterative Residual Correction.- References.- Defect Corrections in Applied Mathematics and Numerical Software.- Defect Corrections and Hartree-Fock Method.- Hartree-Fock Method.- Defect Corrections on Infinite Intervals.- Asymptotic Boundary Conditions and Defect Corrections with Changing Boundary Points.- References.- Deferred Corrections Software and Its Application to Seismic Ray Tracing.- Discontinuous Interfaces at Known Locations.- PASVA4, Part I.- Discontinuities at Unknown Locations.- PASVA4, Part II. Algebraic Parameters and Conditions.- Three-dimensional Two-point Ray Tracing.- Future Developments.- References.- Numerical Engineering: Experiences in Designing PDE Software with Selfadaptive Variable Step Size/Variable Order Difference Methods.- Estimate of the Truncation Error.- The Error Equation.- The Ordinary BVP.- 2- and 3-D Elliptic PDE's.- The Ordinary IVP.- Parabolic PDE's.- Three Examples.- Concluding Remarks.- References.