Abstract. For the finite element solution of Poisson’s equation, a local a posteriori error estimation based on the Hypercircle method is proposed. Even for the solution of Poisson’s equation without the H² regularity, this method can provide explicit local error estimation. The efficiency of the proposed method is demonstrated by numerical experiments for the boundary value problem of Poisson’s equation defined on the 2D and 3D domains.

Abstract. Lyapunov Exponent is commonly used as a measure of chaos. Chaos Degree is propsed as another measure of chaos based on information theory. The advantage of Chaos Degree is that it is calculable from data. However, there is a difference between Chaos Degree and Lyapunov Exponent and determination of chaos requires caution by using Chaos Degree. This paper shows information-theoretic interpretation of the difference and proposes improvement of Chaos Degree in consideration of the difference.

Abstract. Floating-point number and its arithmetic are widely used in numerical computations. Because information of the floating-point numbers is finite, a rounding error may occur in floating-point arithmetic. The optimal relative error bound for floating-point arithmetic is given by Jeannerod and Rump. In this paper, we extend the discussion to signed relative error and derive the optimal range of the signed relative error.