Recent Development on Aitken Schwarz Method

The idea of using Aitken acceleration [Hen64] [SB80], on the classical Schwarz additive domain decomposition method has been introduced in [GTD99]. For an elliptic operator with constant coefficient on a regular grid, this method is called Aitken-Schwarz procedure, and is a direct solver. This method has shown very good numerical performances, and has been used in more complex situations [GTD01]. In this work, we extend Aitken-Schwarz procedure to the case of a 2-D cartesian grid, not necessarily regular. The key idea is the replacement of the 1-D Fourier transform used on the regular space step discretization of the artificial interface grid by a transform using the eigenvectors of a suitable 1-D operator. For simplicity, this presentation is limited here to the Laplacian operator and to two subdomains. However, our method can be applied to the Helmholtz operator for example and one-dimensional domain decomposition with an arbitrary number of subdomains. In section 2, we recall the basic idea of Aitken-Schwarz method on a regular grid. In section 3, we describe two extensions of the method on a general cartesian grid : one using all the eigenvectors, the other a limited number of them. Numerical experiments are described and analyzed in section 4.