Yamamotoʹs principle and its applications to precise finite element error analysis

Suppose that we discretize an elliptic boundary value problem and obtain a linear equation Ax=f. In many case, the inverse matrix A−1 is closely related to the Green function of the original boundary value problem. This fact is called Yamamotoʹs principle. In this paper, using Yamamotoʹs principle, we develop a precise error analysis of the piecewise linear finite element method for two-point boundary value problems with discontinuous and not necessarily positive coefficient functions. We show that precise error estimations, similar to known error bounds, are obtained even in this case.