Computable Pointwise Error Bounds and the Ritz Method in One Dimension

This paper is concerned with a posteriors error estimation for the numerical approximation, by the Ritz method, of linear, selfadjoint, two-point boundary value problems. The point of departure is a basic inequality, which is the best possible (given the information used in computing the approximation), that bounds the pointwise error in the Ritz approximation. This optimal bound involves some readily available information and some quantities (functionals of the true solution and Green’s function) that are not, in general, known. The problem of estimating these latter quantities, and thus producing computable bounds, is approached from two different points of view; first, using complementary variational principles, and second, approximating the energy inner product and using the theory of kernel functions. Numerical results are given, and possible extensions to higher dimensions are commented on.