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This book is dedicated to Olivier Pironneau. For more than 250 years partial differential equations have been clearly the most important tool available to mankind in order to understand a large variety of phenomena, natural at first and then those originating from human activity and technological development. Mechanics, physics and their engineering applications were the first to benefit from the impact of partial differential equations on modeling and design, but a little less than a century ago the Schrödinger equation was the key opening the door to the application of partial differential equations to quantum chemistry, for small atomic and molecular systems at first, but then for systems of fast growing complexity. Mathematical modeling methods based on partial differential equations form an important part of contemporary science and are widely used in engineering and scientific applications. In this book several experts in this field present their latest results and discuss trends in the numerical analysis of partial differential equations. The first part is devoted to discontinuous Galerkin and mixed finite element methods, both methodologies of fast growing popularity. They are applied to a variety of linear and nonlinear problems, including the Stokes problem from fluid mechanics and fully nonlinear elliptic equations of the Monge-Ampère type. Numerical methods for linear and nonlinear hyperbolic problems are discussed in the second part. The third part is concerned with domain decomposition methods, with applications to scattering problems for wave models and to electronic structure computations. The next part is devoted to the numerical simulation of problems in fluid mechanics that involve free surfaces and moving boundaries. The finite difference solution of a problem from spectral geometry has also been included in this part. Inverse problems are known to be efficient models used in geology, medicine, mechanics and many other natural sciences. New results in this field are presented in the fifth part. The final part of the book is addressed to another rapidly developing area in applied mathematics, namely, financial mathematics. The reader will find in this final part of the volume, recent results concerning the simulation of finance related processes modeled by parabolic variational inequalities.

Differential equations, Partial. --- Engineering mathematics. --- Physics. --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Engineering --- Engineering analysis --- Mathematical analysis --- Mathematics --- Partial differential equations --- Differential equations, partial. --- Numerical and Computational Physics, Simulation. --- Mathematical and Computational Engineering. --- Partial Differential Equations. --- Mathematical Modeling and Industrial Mathematics. --- Applied mathematics. --- Partial differential equations. --- Mathematical models. --- Models, Mathematical --- Simulation methods --- Modeling --- Numerical simulation

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Recent decades have seen a very rapid success in developing numerical methods based on explicit control over approximation errors. It may be said that nowadays a new direction is forming in numerical analysis, the main goal of which is to develop methods ofreliable computations. In general, a reliable numerical method must solve two basic problems: (a) generate a sequence of approximations that converges to a solution and (b) verify the accuracy of these approximations. A computer code for such a method must consist of two respective blocks: solver and checker.In this book, we are chie

Numerical analysis. --- Error-correcting codes (Information theory) --- Approximation theory. --- Theory of approximation --- Functional analysis --- Functions --- Polynomials --- Chebyshev systems --- Codes, Error-correcting (Information theory) --- Error-detecting codes (Information theory) --- Forbidden-combination check (Information theory) --- Self-checking codes (Information theory) --- Artificial intelligence --- Automatic control --- Coding theory --- Information theory --- Mathematical analysis

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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.   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.

Computer science. --- Numerical analysis. --- Computer mathematics. --- Physics. --- Computational intelligence. --- Computer Science. --- Numeric Computing. --- Computational Science and Engineering. --- Numerical Analysis. --- Numerical and Computational Physics. --- Computational Intelligence. --- Mathematical analysis --- Intelligence, Computational --- Artificial intelligence --- Soft computing --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Computer mathematics --- Discrete mathematics --- Electronic data processing --- Informatics --- Science --- Mathematics --- Electronic data processing. --- Engineering. --- Numerical and Computational Physics, Simulation. --- Construction --- Industrial arts --- Technology --- ADP (Data processing) --- Automatic data processing --- Data processing --- EDP (Data processing) --- IDP (Data processing) --- Integrated data processing --- Computers --- Office practice --- Automation --- Numerical calculations --- Mathematical models. --- Verification.

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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.   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.

Numerical analysis --- Mathematical physics --- Applied physical engineering --- Computer science --- Information systems --- Artificial intelligence. Robotics. Simulation. Graphics --- Computer. Automation --- neuronale netwerken --- fuzzy logic --- cybernetica --- theoretische fysica --- computers --- informatica --- informaticaonderzoek --- KI (kunstmatige intelligentie) --- ingenieurswetenschappen --- computerkunde --- robots --- numerieke analyse --- gegevensverwerking --- AI (artificiële intelligentie) --- Numerical calculations --- Mathematical models. --- Verification.