On the asymptotic convergence of collocation methods

Abstract

Abstract. We prove quasioptimal and optimal order estimates in various Sobolev norms for the approximation of linear strongly elliptic pseudodifferential equations in one independent variable by the method of nodal collocation by odd degree polynomial splines. The analysis pertains in particular to many of the boundary element methods used for numerical computation in engineering applications. Equations to which the analysis is applied include Fredholm integral equations of the second kind, certain first kind Fredholm equations, singular integral equations involving Cauchy kernels, a variety of integro-differential equations, and two-point boundary value problems for ordinary differential equations. The error analysis is based on an equivalence which we establish between the collocation methods and certain nonstandard Galerkin methods. We compare the collocation method with a standard Galerkin method using splines of the same degree, showing that the Galerkin method is quasioptimal in a Sobolev space of lower index and furnishes optimal order approximation for a range of Sobolev indices containing and extending below that for the collocation method, and so the standard Galerkin method achieves higher rates of convergence.1. Introduction. In this paper we apply the method of nodal collocation by odd degree polynomial splines to systems of strongly elliptic pseudodifferential equations on closed curves and to two-point boundary value problems for ordinary differential equations. (By splines we always mean smoothest splines.) The former class of problems encompasses many of the boundary integral equations discretized by boundary element methods including some first kind and second kind Fredholm integral equations, singular integral equations with Cauchy kernels, and certain integro-differential equations such as the normal derivative of the double layer potential and the operator of Prandtl's wing theory.For spline approximation of such strongly elliptic equations via standard Galerkin procedures or via the corresponding numerically integrated Galerkin-collocation methods the asymptotic error analysis is already rather completely developed; see

Pointwise convergence of some boundary element methods. Part II

“…for 2a-and max {<*,ƒ>} ^q^k ; see [4], [17], [18], [20]. Using the inverse inequality for S^m in the usual manner, the mean square result (2.7) also gives the pointwise error estimate (see also [17])…”

“…The naive spline collocation with odd degree splines for boundary element methods in two dimensions, n = 2, has been analyzed in [4] as a modified Galerkin method. Therefore, our results on the pointwise error estimâtes can be applied to that modification.…”

“…Let now F in IR 2 be a finite system of mutually disjoint Jordan curves Tj, where as in [4], every F ; can be considered to be the smooth image of a corresponding homeomorphism of the unit circle which is parametrized by v = (cos 2 Trt, sin 2 irt ) T . Then, the original équation (1.1) is equivalent to a system of équations of the form Furthermore, let A be a strongly elliptic system of pseudo-differential operators of degree 2 a, Le., corresponding to A there exists a regular smooth matrix-valued function 6 on F such that Note that this concept of strong ellipticity is weaker than the coercivity assumption (1.2), (1.3) for the Galerkin method (see also [4], Section 2.3).…”

“…Then, the original équation (1.1) is equivalent to a system of équations of the form Furthermore, let A be a strongly elliptic system of pseudo-differential operators of degree 2 a, Le., corresponding to A there exists a regular smooth matrix-valued function 6 on F such that Note that this concept of strong ellipticity is weaker than the coercivity assumption (1.2), (1.3) for the Galerkin method (see also [4], Section 2.3).…”

A practical examination of the errors arising in the direct collocation boundary element method for acoustic scattering

“…In an academic sense, formal mathematical investigations on the convergence of the various discretisation and numerical procedures lie within the (now) historical literature (e.g., [1,2]). In contrast, within engineering fields it is recurrently only the relative number of elements that is considered, with six elements per wavelength the most frequently prescribed guideline (see [3] and references therein).…”