A GENERALIZATION OF THE ARNOLD-WENDLAND LEMMA TO A MODIFIED COLLOCATION METHOD FOR BOUNDARY INTEGRAL-EQUATIONS IN R(3)

We prove quasioptimal and optimal order estimates in various Sobolev n orms for the approximation of linear strongly elliptic periodic pseudo differential equations in two independent variables by a modified meth od of nodal collocation by odd degree polynomial splines. In the one-d imensional case, our method coincides with the method of nodal colloca tion when odd degree polynomial splines are employed for the trial fun ctions. The convergence analysis is based on an equivalence which we e stablish between our method and a nonstandard Galerkin method for an o perator closely related to the given operator. This equivalence is rea lized through a crucial intermediate result (which we now term the Arn old-Wendland lemma) to connect the solution of central finite differen ce equations and that of certain nonstandard Galerkin equations. The r esults of this paper are genuine two-dimensional generalizations of th e results obtained by ARNOLD and WENDLAND in [2] for the one-dimension al equations.