HIGH-ORDER SPLINE PETROV-GALERKIN METHODS WITH QUADRATURE

This note is concerned with a high-order spline Petrov-Galerkin method for m-th order two-point boundary value problems and generalizations in which the L(2)-inner product is replaced by a composite quadrature rule. The elementary rule from which the composite rule is formed need not be accurate, but its weights are required to be positive, and the number of points cannot be smaller than a certain minimum value. Cert ain collocation methods are included as a special case. There are no r estrictions on mesh ratios. The stability and hence convergence of the methods in L(p)-spaces depends on the uniform boundedness of a discre te analogue of the L(2) orthogonal projection on certain spline spaces with respect to the L(p)-norm. Under additional assumptions on the qu adrature rules, superconvergence results for the approximate solution and eigenvalues can be proved.