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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 the authors' methods. Also in some
examples practical computer algorithms are supplied so that readers
can easily implement the verification programs by themselves.
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 the authors' methods. Also in some
examples practical computer algorithms are supplied so that readers
can easily implement the verification programs by themselves.
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