# Numerical verifications for solutions to elliptic equations using residual iterations with a higher order finite element

@article{Yamamoto1995NumericalVF, title={Numerical verifications for solutions to elliptic equations using residual iterations with a higher order finite element}, author={Nobito Yamamoto and Mitsuhiro T. Nakao}, journal={Journal of Computational and Applied Mathematics}, year={1995}, volume={60}, pages={271-279} }

## 57 Citations

### Numerical Verification of Solutions for Nonlinear Elliptic Problems Using anL∞Residual Method☆

- Mathematics
- 1998

Abstract We consider a numerical enclosure method with guaranteedL∞error bounds for the solution of nonlinear elliptic problems of second order. By using an a posteriori error estimate for the…

### Numerical verification methods for solutions of semilinear elliptic boundary value problems

- Mathematics
- 2011

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…

### A numerical verification method for solutions of singularly perturbed problems with nonlinearity

- Mathematics
- 2005

In order to verify the solutions of nonlinear boundary value problems by Nakao’s computerassisted numerical method, it is required to find a constant, as sharp as possible, in the a priori error…

### A Numerical Method to Verify the Invertibility of Linear Elliptic Operators with Applications to Nonlinear Problems

- MathematicsComputing
- 2004

A numerical method to verify the invertibility of second-order linear elliptic operators by using the projection and the constructive a priori error estimates based upon the existing verification method originally developed by one of the authors is proposed.

### An approach to the numerical verification of solutions for nonlinear elliptic problems with local uniqueness

- Mathematics
- 1999

We propose a numerical method to verify the existence and local uniqueness of solutions to nonlinear elliptic equations. We numerically construct a set containing solutions which satisfies the…

### Numerical verifications for eigenvalues of second-order elliptic operators

- Mathematics
- 1999

In this paper, we consider a numerical technique to verify the exact eigenvalues and eigenfunctions of second-order elliptic operators in some neighborhood of their approximations. This technique is…

### A simple numerical verification method for differential equations based on infinite dimensional sequential iteration

- Mathematics
- 2013

This paper describes a numerical verification of solutions for infinite dimensional functional equations based on residual form and sequential iteration. Comparing with other verification procedures…

### Numerical verification methods for a system of elliptic PDEs, and their software library

- Computer Science, MathematicsNonlinear Theory and Its Applications, IEICE
- 2021

Existing verification methods are reformulated using a convergence theorem for simplified Newton-like methods in the direct product space of a computable finite-dimensional space Vh and its orthogonal complement space V⊥ to provide verification methods of solutions to PDEs.

### NUMERICAL VERIFICATION METHODS FOR SOLUTIONS OF ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS

- Mathematics
- 2000

In this article, we describe on a state of the art of validated numerical computations for solutions of differential equations. A brief overview of the main techniques for self-validating numerics…

### A residual bound evaluation of operator equations with Raviart-Thomas finite element (The latest developments in theory and application on scientific computation)

- Mathematics
- 2012

Abstract –In this article, a residual evaluation of operator equation is considered in the framework of computer-assisted proof. Our computer-assisted approach ensures the existence and local…

## References

SHOWING 1-10 OF 12 REFERENCES

### Solving Nonlinear Elliptic Problems with Result Verification Using an H -1 Type Residual Iteration

- Mathematics
- 1993

Solving Nonlinear Elliptic Problems with Result Verification Using an H -1 Type Residual Iteration. In this paper, we consider a numerical technique to verify the solutions with guaranteed error…

### Numerical verifications of solutions for nonlinear elliptic equations

- Mathematics
- 1993

A numerical technique which enables us to verify the existence of weak solutions for nonlinear elliptic boundary value problems is proposed. It is based on the infinite dimensional fixed point…

### A numerical verification method for the existence of weak solutions for nonlinear boundary value problems

- Mathematics
- 1992

### A Computational Verification Method of Existence of Solutions for Nonlinear Elliptic Equations

- Mathematics
- 1989

### Numerical verifications of solutions for elliptic equations with strong nonlinearity

- Mathematics
- 1991

Numerical methods for automatic proof of the existence and the local uniqueness of weak solutions of elliptic boundary value problems with strongly nonlinear terms are proposed. They are based on the…

### On computational proofs of the existence of solutions to nonlinear parabolic problems

- Mathematics
- 1994

### Solving nonlinear parabolic problems with result verification Part I: one-space dimensional case

- Mathematics
- 1991

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

- Mathematics
- 1988

This paper is a continuation of the preceding study ([2]) in which we described an automatic proof by computer, utilizing Schauder’s fixed point theorem, of the existence of weak solutions for…

### Numerical verifications of solutions for elliptic equations in nonconvex polygonal domains

- Mathematics
- 1993

SummaryIn this paper, methods for numerical verifications of solutions for elliptic equations in nonconvex polygonal domains are studied. In order to verify solutions using computer, it is necessary…