On very accurate verification of solutions for boundary value problems by using spectral methods

Abstract

In this paper, we consider a numerical verification method of solutions for nonlinear elliptic boundary value problems with very high accuracy. We derive a constructive error estimates for the $H^1_0$-projection into polynomial spaces by using the property of the Legendre polynomials. On the other hand, the Galerkin approximation with higher degree polynomials enables us to get very small residual errors. Combining these results with existing verification procedures, several verification examples which confirm us the actual effectiveness of the method are presented.