Domain decomposition for an advection--diffusion problem with parabolic layers.
The authors consider a singularly perturbed convection-diffusion equation in the unit square with the convection from the right to the left. The essential features of the solution are an exponential boundary layer at the left boundary and parabolic boundary layers at the lower and the upper boundary. The equation is discretized by a finite difference upwind scheme using either a Shishkin mesh which is adapted to the exponential layer or a Shishkin mesh which is adapted to the parabolic layers. The numerical solution of the discrete system is performed with domain decomposition methods. In the first method, the domain is decomposed into vertical strips (parallel to the exponential layer) and in the second method into horizontal strips (parallel to the parabolic layers). The main feature of these methods is the use of auxiliary domains which cover the interfaces of the subdomains. In the algorithm, local problems with Dirichlet boundary conditions are solved alternating on the subdomains and on the auxiliary domains. The convergence of this domain decomposition method is proved. The rate of convergence is estimated and it is shown that under certain conditions this rate is independent of the perturbation parameter. Finally, the paper presents numerical tests which confirm the analytical results. In these tests, the number of iterations for the decomposition into horizontal strips was much smaller than for the vertical decomposition.
scientific article; zbMATH DE number 2009018