Optimized Schwarz methods for the advection-diffusion equation
Dubois, Olivier · 2003
The optimized Schwarz methods were recently introduced to enhance the convergence of the classical Schwarz iteration, by replacing the Dirichlet transmission conditions with different conditions obtained through an optimization of the convergence rate. This is formulated as a min-max problem. These new methods are well-studied for elliptic second order symmetric equations. The purpose of this work is to compute optimized Robin transmission conditions for the advection-diffusion equation in two dimensions, by finding the solution of the min-max problem. The asymptotic expansion, for small mesh size h, of the resulting convergence rate is found: it shows a weak dependence on h, if the overlap is 0(h) or no overlap is used. Numerical experiments illustrate the improved convergence of these optimized methods compared to other Schwarz methods, and also justify the continuous Fourier analysis performed on a simple model problem only. The theoretical asymptotic performance is also verified numerically.