Elliptic spline functions and the Rayleigh-Ritz-Galerkin method

Error estimates for the Rayleigh-Ritz-Galerkin method, using finite-dimensional spline type spaces, for a class of nonlinear two-point boundary value problems are dis- cussed. The results of this paper extend and improve recent corresponding results of B. L. Hulme, F. M. Perrin, H. S. Price, and R. S. Varga. 1. Introduction. The purpose of this paper is to discuss error bounds for the Rayleigh-Ritz-Galerkin method, using finite-dimensional "spline-type" spaces, for a class of nonlinear two-point boundary value problems, cf. (2), (3), (4), (5), (7), and (9). In particular, we generalize, extend, and simplify the very important techniques and results of (7), (9), and (10). In Section 2, we generalize the interpolation theory results of (11) and (12) to spline spaces defined by an arbitrary selfadjoint, elliptic, ordinary differential operator and in Section 3 we analyze and apply these results to the class of non- linear two-point boundary value problems previously studied extensively in (3) and (4). Now we introduce some notations, which will be used throughout this paper. Let a and b be two fixed real numbers such that -oo < a < b < oo. If u E C- (a, b) and is real-valued, for each nonnegative integer m and 1 ? p < co, let (/b m \/p d llullmp E Z Dju(x)JPdx , whereD- ' j~~~~~o ~~~dx' WMIp denote the closure of the set {u E C-(a, b), u real-valued I Ijulm,p < co } with respect to 11 1m.P and WomP denote the closure of the real-valued functions in Co-(a, b), i.e., the real-valued C-(a, b)-functions with compact support contained in the interior of (a, b), with respect to 11 imP. We remark that u E WmP if and only if u E Cm-l(a, b), Dm-lu is absolutely continuous, and Dmu E LP(a, b). Moreover, u E WomP if and only if u E WmP and Dku(a) = Dku(b) = 0 0 < k _ m - 1. Finally, the symbol K will be used repeatedly to denote a positive constant, not necessarily the same at each occurrence.