ON THE ASYMPTOTIC CONVERGENCE OF SPLINE COLLOCATION METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS.

The asymptotic accuracy of the method of collocation for the approximate solution of linear elliptic partial differential equations is examined. Specifically, the authors consider the nodal collocation of a second order equation in the plane with biperiodicity conditions using tensor product smooth splines of odd degree as trial functions. They prove optimal rates of convergence in L**2 for partial derivatives of the approximate solution which are of order at least two in one variable, while the solution itself and its gradient converge in L**2 at rates less than the optimal approximation theoretic results.