A method of computer assisted proof for nonlinear two-point boundary value problems using higher order finite elements

Abstract

Present authors have presented with Takayuki Kubo at University of Tsukuba a method of a computer assisted proof for the existence and uniqueness of solutions to two-point boundary value problems of nonlinear ordinary differential equations in the paper submitted for NOLTA, IEICE. This method uses piecewise linear finite element base functions and sometimes requires fine mesh. To overcome this difficulty, in this paper, an improved method is presented for the norm estimation of the residual to the operator equation. In this refined formulation, piecewise quadratic finite element base functions are used. A kind of the residual technique works sophisticatedly well. It is stated that the estimation of the residual can be expected smaller than that of the previous method. Finally, four examples are presented. Each result demonstrates that a remarkable improvement is achieved in accuracy of the guaranteed error estimation.