On the asymptotic convergence of collocation methods

Douglas N. Arnold, Wolfgang L. Wendland

Research output: Contribution to journalArticlepeer-review

74 Scopus citations

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.

Original languageEnglish (US)
Pages (from-to)349-381
Number of pages33
JournalMathematics of Computation
Volume41
Issue number164
DOIs
StatePublished - Oct 1983

Fingerprint Dive into the research topics of 'On the asymptotic convergence of collocation methods'. Together they form a unique fingerprint.

Cite this