Local Piecewise Polynomial Projection Methods

Local projection methods which yield c'm_1) piecewise polynomials of order m + k as approximate solutions of a boundary value problem for an mth order ordinary differential equation are determined by the k linear functional at which the residual error in each partition interval is required to vanish on. We develop a condition on these k f unctionals which implies breakpoint superconvergence (of derivatives of order less Üian m) for the approximating piecewise polynomials. The same order of superconvergence is associated with eigenvalue problems. A discrete connection between two particular projectors yielding 6(|A|2*) superconvergence, namely (a) collocation at the k Gauss-Legendre points in each partition interval and (b) "essential least-squares" (i.e., local moment methods), is made by asking that this same order of superconvergence result when using collocation at k — r points per interval and simultaneous local orthogonality of the residual to polynomials of order r; the k — r points then necessarily form a subset of the k Gauss-Legendre points. Introduction. This is the last in a triple (see [2], [3]) of papers concerned with high-order approximation to eigenvalues of an O.D.E. using collocation at Gauss points. Correspondingly, its two sections are labelled 9 and 10, but it can be read without reference to [3], i.e., to Sections 5-8. Items labelled x.y or (x.y) are to be found in Section x, e.g., in [2] in case x is less than 5. When writing [2], we were forced to go through the arguments in [1] once again and ended up improving upon them somewhat; see the proof of Theorem 9.2 below. In the process, we considered more general local piecewise polynomial projection methods in an effort to discover just what produces the superconvergence at breakpoints in Gauss-point collocation. This led us to a simple set of conditions on the local projector used which, so we found, had been formulated much earlier by Pruess [4] in another context. In addition to updating our earlier results in [1] and [2] to cover this wider class of projection methods, we give a detailed analysis of these special local projectors and establish a simple link between the two best known among these, viz. Interpolation at Gauss points and Least-squares approximation. 9. Some Projectors Which Yield Superconvergence. As de Boor and Swartz [1] describe it, local projection methods which involve sufficiently rough piecewise Received April 11, 1980. 1980 Mathematics Subject Classification. Primary 65L15, 65Jxx.