Kouji Hashimoto

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

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 simplify the verification process. Several numerical examples that confirm the actual effectiveness of the method are presented.

Numerical validation of solutions of saddle point matrix equations

A numerical validation method for verifying the accuracy of approximate solutions of saddle point matrix equations is presented and analysed. The method only requires iterative solutions of two symmetric positive definite linear systems. Moreover, it is shown that preconditioning can be used to improve the error bounds. The method is illustrated by several examples derived from mixed finite element discretization of the Stokes equations. Preliminary numerical results indicate that the method is efficient. Copyright

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

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 estimates for the finite element approximation of some simple linear problems. For singularly perturbed problems, however, generally it is known that the perturbation term produces a bad effect on the a priori error estimates, i.e., leads to a large constant, if we use the usual approximation methods. In this paper, we propose some verification algorithms for solutions of singularly perturbed problems with nonlinearity by using the constant obtained in the a priori error estimates based on the exponential fitting method with Green’s function. Some numerical examples which confirm us the effectiveness of our method are presented.

Numerical Verification Methods for Solutions of the Free Boundary Problem

ABSTRACT We propose two methods to enclose the solution of an ordinary free boundary problem. The problem is reformulated as a nonlinear boundary value problem on a fixed interval including an unknown parameter. By appropriately setting a functional space that depends on the finite element approximation, the solution is represented as a fixed point of a compact map. Then, by using the finite element projection with constructive error estimates, a Newton-type verification procedure is derived. In addition, numerical examples confirming the effectiveness of current methods are given.

Verification methods for nonlinear equations with saddle point functions

This paper presents a fast verification algorithm for nonlinear equations with saddle point functions. This algorithm is based on a block decomposition of the Krawczyk-type interval operator, which can be applied to convex programming problems and nonlinear Navier-Stokes equations. We show the efficiency of this algorithm by comparing it with the Krawczyk method and the interval Newton-like method for the discretized stationary Navier-Stokes equations.