A Numerical Method to Verify the Invertibility of Linear Elliptic Operators with Applications to Nonlinear Problems
@article{Nakao2004ANM, title={A Numerical Method to Verify the Invertibility of Linear Elliptic Operators with Applications to Nonlinear Problems}, author={Mitsuhiro T. Nakao and Kouji Hashimoto and Yoshitaka Watanabe}, journal={Computing}, year={2004}, volume={75}, pages={1-14} }
Abstract.In this paper, we propose a numerical method to verify the invertibility of second-order linear elliptic operators. By using the projection and the constructive a priori error estimates, the invertibility condition is formulated as a numerical inequality based upon the existing verification method originally developed by one of the authors. As a useful application of the result, we present a new verification method of solutions for nonlinear elliptic problems, which enables us to…
81 Citations
An improvement of the theorem of a posteriori estimates for inverse elliptic operators
- Mathematics
- 2014
: This paper presents a numerical method to verify the invertibility of a linear elliptic operator. The invertibility of a linearized operator is useful information when verifying the existence of a…
An improved method for verifying the existence and bounds of the inverse of second-order linear elliptic operators mapping to dual space
- MathematicsJapan Journal of Industrial and Applied Mathematics
- 2019
This paper presents an improved method for determining the invertibility of second-order linear elliptic operators with a bound on the norm of their inverses by computers in a mathematically rigorous…
Verified norm estimation for the inverse of linear elliptic operators using eigenvalue evaluation
- Mathematics
- 2014
This paper proposes a verified numerical method of proving the invertibility of linear elliptic operators. This method also provides a verified norm estimation for the inverse operators. This type of…
Some considerations of the invertibility verifications for linear elliptic operators
- Mathematics
- 2015
This paper presents three computer-assisted procedures for verifying the invertibility of second-order linear elliptic operators and for computing a bound on the norm of its inverse. One of these…
Some improvements of invertibility verifications for second-order linear elliptic operators
- Mathematics
- 2020
Some Remarks on the Rigorous Estimation of Inverse Linear Elliptic Operators
- Mathematics, Computer ScienceSCAN
- 2014
A new numerical method is presented to obtain the rigorous upper bounds of inverse linear elliptic elliptic operators and it is shown the proposed new estimate is effective for an intermediate mesh size.
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…
On very accurate verification of solutions for boundary value problems by using spectral methods
- MathematicsJSIAM Lett.
- 2009
A constructive error estimates for the H 0 -projection into polynomial spaces are derived by using the property of the Legendre polynomials and the Galerkin approximation with higher degree polynoms enables us to get very small residual errors.
A numerical verification method for solutions of nonlinear parabolic problems
- Mathematics
- 2009
By using the finite element approximation and constructive a priori error estimates, a new formulation for proving the existence of solutions for nonlinear parabolic problems is presented. We present…
On a posteriori estimates of inverse operators for linear parabolic initial-boundary value problems
- MathematicsComputing
- 2011
A new technique for obtaining the estimates of the inverse operator by using the finite dimensional approximation and error estimates enables us to obtain very sharp bounds compared with a priori estimates.
References
SHOWING 1-10 OF 22 REFERENCES
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 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…
A numerical approach to the proof of existence of solutions for elliptic problems
- Mathematics
- 1988
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…
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 for solutions to elliptic equations using residual iterations with a higher order finite element
- Mathematics
- 1995
An Efficient Approach to the Numerical Verification for Solutions of Elliptic Differential Equations
- MathematicsNumerical Algorithms
- 2004
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.
A Numerical Verification Method for Solutions of Boundary Value Problems with Local Uniqueness by Banach's Fixed-Point Theorem
- Mathematics
- 1998
In this paper, we propose a method to prove the existence and the local uniqueness of solutions to infinite-dimensional fixed-point equations using computers. Choosing a set which possibly includes a…
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 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…