Aitken-Schwarz Acceleration not based on the mesh for CFD

Abstract

The generalized Schwarz alternating method (GSAM) was introduced by [4]. Its purely linear convergence in the case of linear operators suggests that the convergent sequence of trace solutions at the artificial interfaces can be accelerated by the well known process of Aitken convergence acceleration. This is the basis of the Aitken-Schwarz method proposed by [1]. In [3] authors extend the Aitken acceleration method to nonuniform meshes, by developing a new original method to compute the Non Uniform Discrete Fourier Transform (NUDFT) based on the function values at the nonuniform points. Nevertheless, the acceleration used was based on the mesh, as it requires an orthogonal basis related to this mesh to decompose the iterated solution at the artificial interfaces. In this paper, we develop a technique that have the same benefits as the Fourier transform: an orthogonal basis to represent the iterated solution and a decrease of the coefficient of the solution related to this basis. This technique creates a robust framework for the adaptive acceleration of the Schwarz method, by using an approximation of the error operator at artificial interfaces based on a posteriori estimate of the modes behavior.