A numerical approach to the proof of existence of solutions for elliptic problems

@article{Nakao1988ANA,
  title={A numerical approach to the proof of existence of solutions for elliptic problems},
  author={Mitsuhiro T. Nakao},
  journal={Japan Journal of Applied Mathematics},
  year={1988},
  volume={5},
  pages={313-332}
}
  • M. Nakao
  • Published 1 November 1988
  • Mathematics
  • Japan Journal of Applied Mathematics
In this paper, we describe a method which proves by computers the existence of weak solutions for linear elliptic boundary value problems of second order. It is shown that we can constitute the computing procedures to verify the existence, uniqueness and inclusion set of a solution based on Schauder’s fixed point theorem. Using the finite element approximations for some simple Poisson’s equations and the results of error estimates, we generate iteratively a set sequence composed of functions… 

Computer assisted proofs of solutions to Nonlinear elliptic partial di ff erential equations

In this article, a numerical method is presented for computer assisted proofs to the existence and uniqueness of solutions to Dirichlet boundary value problems in a certain class of nonlinear

Numerical Verifications of Solutions for Nonlinear Hyperbolic Equations

In this paper, we consider a numerical technique to enclose the solutions with guaranteed error bounds for nonlinear hyperbolic initial boundary value problems as well as to verify the existence of

A Method of Verified Computations for Solutions to Semilinear Parabolic Equations Using Semigroup Theory

TLDR
A recursive scheme to extend a time interval in which the validity of the solution can be verified and the existence of a global-in-time solution is demonstrated for a certain semilinear parabolic equation.

An Efficient Approach to the Numerical Verification for Solutions of Elliptic Differential Equations

TLDR
An alternative method to overcome the difficulty of verifying the accuracy of numerical verification methods for solutions of second-order elliptic boundary value problems based on the infinite-dimensional fixed-point theorem is proposed.

Numerical verification methods for solutions of semilinear elliptic boundary value problems

This article describes a survey on numerical verification methods for second-order semilinear elliptic boundary value problems introduced by authors and their colleagues. Here “numerical

Numerical verification method for positive solutions of elliptic problems

A COMPUTATIONAL VERIFICATION METHOD OF SOLUTION WITH UNIQUENESS FOR OBSTACLE PROBLEMS

A numerical method for automatic proof of the existence of solutions for variational inequalities is proposed. It is based on the infinite di mensional fixed point theorem and computable error

Numerical Verification Method of Solutions for Elliptic Variational Inequalities

In this chapter, we propose numerical techniques which enable us to verify the existence of solutions for the free boundary problems governed by two kinds of elliptic variational inequalities. Based

Numerical Existence Proofs and Guaranteed Error Bounds for Solutions to Two-Point Boundary Value Problems (Recent Developments of Numerical Analysis and Numerical Computation Algorithms)

Abstract –In this article, a numerical method is presented for verifying the existence and the uniqueness of solutions to two-point boundary value problems of second order ordinary differential

On verified computations of solutions for nonlinear parabolic problems

: We consider the methods for guaranteed computations of solutions for nonlinear parabolic initial-boundary value problems. First, in order to make the basic principle clear, we briefly introduce the
...