On the Asymptotic Convergence of Collocation Methods With Spline Functions of Even Degree

Abstract

Abstract. We investigate the collocation of linear one-dimensional strongly elliptic integro-differential 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 ß-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 variational 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 principal 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.

Einführung in moderne Galerkin-Randeiementmethoden mit einer Anwendung aus dem Maschinenbau

“…Mit Hilfe dieser Formulierung wurde erstmals rigoros die asymptotische Konvergenz des Kollokafionsverfahrens mit Splines ungeraden Grades ffr die allgemeine Klasse der stark elliptischen Pseudodifferentialgleichungen bewiesen, wobei gegeniiber der Arbeit yon Pr6ssdorf/Schmidt [104] unregelm~iftige Netze betrachtet wurden. Ffr eindimensionale Integralgleichungen ist die Konvergenz der N~herungsl6sung gefen die exakte L6sung also beispielsweise beim Kollokationsverfahren abgesichert [2,3,113,112], und in der Literatur werden spezietle Ansatzfunktionen mit globalem Tr/iger angegeben, die geeigneter sind als die stfickweisen Polynomans~itze, die in der Randelementmethode verwendet werden. Eine grofe Anzahl yon weiteren Resultaten und viele Referenzen zu diesem Thema sind in dem Lehrbuch yon Pr6ssdorf/Silbermann [106] …”

“…,(113) und(114) verknfipft sind, definiert werden durch Vij*i := 2 ;4 x fr "/~,yU(X, y -x)*~(y)ds(y) ,Koq~ i := 2 v.p. YJo~ 'F~.yT(x, r -x)qoi(y)ds(( x, r-x)m~(y)ds(r) , wobei B~(x) = {), : ]y -x] < s (e > 0)} die e-Kugel mit dem Mittelpunkt x E N~ und dem Radius e ist.…”

Spline collocation for convolutional parabolic boundary integral equations

“…To our knowledge, there are results only for some special cases, where the equation is given on a torus [3,32,9], square or cube [11,31]. For the spline collocation in the one dimensional case we refer to the basic papers [4,5,33,34]. In our case the integral equation is given on a cylindrical domain.…”

An optimal order collocation method for first kind boundary integral equations on polygons

“…to demonstrate that the discretization of the integral operator has a bounded inverse in some appropriate sense. This question is completely answered for the case when Γ is smooth (see [1], [28], [27], [24] and many later papers cited in [30]), but the methods of proof used there do not apply to polygonal Γ . For such Γ the operator on the left hand side of (1.1) can no longer be written as a compact perturbation of a certain well studied isometric isomorphism between H 0 and H 1 .…”

“…The basis functions used are periodic smooth splines of any order k (degree k − 1), with collocation points taken to be mid-points of subintervals when k is odd and break-points when k is even. It is well known (see [1], [28], [24]) that this collocation method is stable for the equation Aw = g (which arises when Γ is a circle). In Theorem 9 of Sect.…”