Quartic spline collocation methods for elliptic partial differential equations

Abstract

This paper discusses quartic spline collocation methods for solving linear second-order elliptic partial differential equations (PDEs). The standard formulation of these methods leads to non-optimal approximations. Here we derive optimal quartic spline approximations, so, high order perturbations of the PDE problem are generated. These perturbations can be applied either to PDE problem operators or to the right sides, thus leading to two different formulations of quartic spline collocation methods. Both methods exhibit optimal order of convergence, that is, optimal O(h5-j) global error estimates for the jth partial derivative are obtained for a certain class of problems. Moreover, O(h6-j) error bounds for the jth partial derivative are obtained at certain sets of points. Numerical experiments show that the quartic spline collocation methods are very efficient from the computational point of view. © 2004 Elsevier Inc. All rights reserved.