Accuracy Verification Methods Theory and Algorithms [Hardcover]

The importance of accuracy verification methods was understood at the very beginning of the development of numerical analysis. Recent decades have seen a rapid growth of results related to adaptive numerical methods and a posteriori estimates. However, in this important area there often exists a noticeable gap between mathematicians creating the theory and researchers developing applied algorithms that could be used in engineering and scientific computations for guaranteed and efficient error control.

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The goals of the book are to (1) give a transparent explanation of the underlying mathematical theory in a style accessible not only to advanced numerical analysts but also to engineers and students; (2) present detailed step-by-step algorithms that follow from a theory; (3) discuss their advantages and drawbacks, areas of applicability, give recommendations and examples.

This book presents recent trends and advances in the theory of directly computable error estimates (error indicators). It discusses the corresponding numerical algorithms in the context of various typical models in engineering and scientific computations.

1 Errors Arising In Computer Simulation Methods.- 1.1 General scheme.- 1.2 Errors of mathematical models.- 1.3 Approximation errors.- 1.4 Numerical errors.- 2 Error Indicators.- 2.1 Error indicators and adaptive numerical methods.- 2.1.1 Error indicators for FEM solutions.- 2.1.2 Accuracy of error indicators.- 2.2 Error indicators for the energy norm.- 2.2.1 Error indicators based on interpolation estimates.- 2.2.2 Error indicators based on approximation of the error functional.- 2.2.3 Error indicators of the Runge type.- 2.3 Error indicators for goal-oriented quantities.- 2.3.1 Error indicators relying on the superconvergence of averaged fluxes in the primal and adjoint problems.- 2.3.2 Error indicators using the superconvergence of approximations in the primal problem.- 2.3.3 lc+