Numerical Verification Methods and Computer-Assisted Proofs for Partial Differential Equations

Michael Plum, Yoshitaka Watanabe, Mitsuhiro T. Nakao

ISBN 10: 9811376719 / ISBN 13: 9789811376719

Editore: Springer Verlag, Singapore, SG, 2020

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In the last decades, various mathematical problems have been solved by computer-assisted proofs, among them the Kepler conjecture, the existence of chaos, the existence of the Lorenz attractor, the famous four-color problem, and more. In many cases, computer-assisted proofs have the remarkable advantage (compared with a "theoretical" proof) of additionally providing accurate quantitative information.The authors have been working more than a quarter century to establish methods for the verified computation of solutions for partial differential equations, mainly for nonlinear elliptic problems of the form -?u=f(x,u,?u) with Dirichlet boundary conditions. Here, by "verified computation" is meant a computer-assisted numerical approach for proving the existence of a solution in a close and explicit neighborhood of an approximate solution. The quantitative information provided by these techniques is also significant from the viewpoint of a posteriori error estimates for approximate solutions of the concerned partial differential equations in a mathematically rigorous sense.In this monograph, the authors give a detailed description of the verified computations and computer-assisted proofs for partial differential equations that they developed. In Part I, the methods mainly studied by the authors Nakao and Watanabe are presented. These methods are based on a finite dimensional projection and constructive a priori error estimates for finite element approximations of the Poisson equation. In Part II, the computer-assisted approaches via eigenvalue bounds developed by the author Plum are explained in detail. The main task of this method consists of establishing eigenvalue bounds for the linearization of the corresponding nonlinear problem at the computed approximate solution. Some brief remarks on other approaches are also given in Part III. Each method in Parts I and II is accompanied by appropriate numerical examples that confirm the actual usefulness of theauthors' methods. Also in some examples practical computer algorithms are supplied so that readers can easily implement the verification programs by themselves.

Codice articolo LU-9789811376719

Dati bibliografici

• Titolo: Numerical Verification Methods and Computer-Assisted Proofs for Partial Differential Equations
• Editore: Springer Verlag, Singapore, SG
• Data di pubblicazione: 2020
• Lingua: Inglese
• ISBN 10: 9811376719
• ISBN 13: 9789811376719
• Rilegatura: Paperback
• Edizione: 2019 ed.
• Condizione: New
• Dimensioni: 15,5 L x 2,77 A x 23,5 P centimetri 15,5 centimetri di larghezza per 2,77 centimetri di altezza per 23,5 centimetri di profondità

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Riassunto

This book is the first publication in the world as a monograph on the concerned research fieldNakao and Plum are pioneers of  the numerical verification method of solution for PDEs
This book offers the basic principle of verification techniques for PDEs as well as interested applications for computer assisted proofs of nonlinear problems

Dalla quarta di copertina

In the last decades, various mathematical problems have been solved by computer-assisted proofs, among them the Kepler conjecture, the existence of chaos, the existence of the Lorenz attractor, the famous four-color problem, and more. In many cases, computer-assisted proofs have the remarkable advantage (compared with a “theoretical” proof) of additionally providing accurate quantitative information.

The authors have been working more than a quarter century to establish methods for the verified computation of solutions for partial differential equations, mainly for nonlinear elliptic problems of the form -?u=f(x,u,?u) with Dirichlet boundary conditions. Here, by “verified computation” is meant a computer-assisted numerical approach for proving the existence of a solution in a close and explicit neighborhood of an approximate solution. The quantitative information provided by these techniques is also significant from the viewpoint of a posteriori error estimates for approximate solutions of the concerned partial differential equations in a mathematically rigorous sense.

In this monograph, the authors give a detailed description of the verified computations and computer-assisted proofs for partial differential equations that they developed. In Part I, the methods mainly studied by the authors Nakao and Watanabe are presented. These methods are based on a finite dimensional projection and constructive a priori error estimates for finite element approximations of the Poisson equation. In Part II, the computer-assisted approaches via eigenvalue bounds developed by the author Plum are explained in detail. The main task of this method consists of establishing eigenvalue bounds for the linearization of the corresponding nonlinear problem at the computed approximate solution. Some brief remarks on other approaches are also given in Part III. Each method in Parts I and II is accompanied by appropriate numerical examples that confirm the actualusefulness of the authors’ methods. Also in some examples practical computer algorithms are supplied so that readers can easily implement the verification programs by themselves.