On the asymptotic convergence of collocation methods with spline functions of even degree

We investigate the collocation of linear one-dimensional strongly elliptic integro-dif- ferential or, more generally, pseudo-differential equations on closed curves by even-degree polynomial splines. The equations are collocated at the respective midpoints subject to uniform nodal grids of the even-degree s-splines. We prove quasioptimal and optimal order asymptotic error estimates in a scale of Sobolev spaces. The results apply, in particular, to boundary element methods used for numerical computations in engineering applications. The equations considered include Fredholm integral equations of the second and the first kind, singular integral equations involving Cauchy kernels, and integro-differential equations having convolutional or constant coefficient principal parts, respectively. The error analysis is based on an equivalence between the collocation and certain varia- tional methods with different degree splines as trial and as test functions. We further need to restrict our operators essentially to pseudo-differential operators having convolutional prin- cipal part. This allows an explicit Fourier analysis of our operators as well as of the spline spaces in terms of trigonometric polynomials providing Babuska's stability condition based on strong ellipticity. Our asymptotic error estimates extend partly those obtained by D. N. Arnold and W. L. Wendland from the case of odd-degree splines to the case of even-degree splines. 1. Introduction. In this paper we investigate the asymptotic convergence of the collocation method using even-degree polynomial splines applied to strongly elliptic systems of pseudo-differential equations on closed curves with convolutional prin- cipal part. The collocation here employs the Gauss points of one-point integration as collocation points which corresponds to the usual boundary element collocation in applications (5), (12), (17), (19), (26), (31), (34). This is in contrast to the method investigated by G. Schmidt in (29) where one collocates at the break points of even-order splines. The asymptotic convergence properties for the standard Galerkin method with splines of arbitrary orders are well-known (22). For the collocation method, however, asymptotic convergence, up to now, has been shown for strongly elliptic systems only in the case of odd-order splines by D. N. Arnold and W. L. Wendland (7).