Convergence of $O(h^4 )$ Cubic Spline Collocation Methods for Elliptic Partial Differential Equations

This paper presents a new class of collocation methods using cubic splines for solving elliptic partial differential equations (PDEs). The error bounds obtained for these methods are optimal. The methods are formulated and a convergence analysis is carried out for a broad class of elliptic PDEs. Experimental results confirm the optimal convergence and indicate that these methods are computationally more efficient than methods based on either collocation with Hermite cubics or on the Galerkin method with cubic splines. Various direct and iterative methods have been applied for the solution of cubic spline collocation methods.