A Generalization of the Arnold-Wendland Lemma to a Modified Collocation Method for Boundary Integral Equations in ℝ3

✍ Scribed by George C. Hsiao; Siegfried Prössdorf

• Publisher: John Wiley and Sons
• Year: 1993
• Tongue: English
• Weight: 531 KB
• Volume: 163
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19931630113

✦ Synopsis

Abstract

We prove quasioptimal and optimal order estimates in various Sobolev norms for the approximation of linear strongly elliptic periodic pseudodifferential equations in two independent variables by a modified method of nodal collocation by odd degree polynomial splines. In the one‐dimensional case, our method coincides with the method of nodal collocation when odd degree polynomial splines are employed for the trial functions. The convergence analysis is based on an equivalence which we establish between our method and a nonstandard Galerkin method for an operator closely related to the given operator. This equivalence is realized through a crucial intermediate result (which we now term the Arnold‐Wendland lemma) to connect the solution of central finite difference equations and that of certain nonstandard Galerkin equations. The results of this paper are genuine two‐dimensional generalizations of the results obtained by ARNOLD and WENDLAND in [2] for the one‐dimensional equations.