Convergence rate of some domain decomposition methods for overlapping and nonoverlapping subdomains (Q1358133)

The problem under consideration is the convection-diffusion equation \[ -\nu \Delta u + a\partial_x u + b\partial_y u + cu = f \] on the entire \((x, y)\)-plane. Here, \(a\), \(b\), \(c\), and \(\nu\) are constants, with \(a\) and \(\nu\) positive and \(c\) non-negative. The domain decomposition involves dividing the plane into a finite number of vertical strips which may or may not overlap. Rates of convergence are obtained for three iterative methods: (1) an additive Schwarz method (a Jacobi-type method), (2) alternate downwind-upwind Gauss-Seidel sweeps, and (3) successive downwind Gauss-Seidel sweeps. The proof is based on results from the algebra of formal languages.