Multigrid methods for cubic spline solution of two point (and 2D) boundary value problems

Abstract

In this paper we propose a scheme based on cubic splines for the solution of the second order two point boundary value problems. The solution of the algebraic system is computed by using optimized multigrid methods. In particular the transformation of the stiffness matrix essentially in a block Toeplitz matrix and its spectral analysis allow to choose smoothers able to reduce error components related to the various frequencies and to obtain an optimal method. The main advantages of our strategy can be listed as follows: (i) a fourth order of accuracy combined with a quadratic conditioning matrix, (ii) a resulting matrix structure whose eigenvalues can be compactly described by a symbol (this information is the key for designing an optimal multigrid method). Finally, some numerics that confirm the predicted behavior of the method are presented and a discussion on the two dimensional case is given, together with few 2D numerical experiments.