The balancing domain decomposition methods by constraints are extended to solving nonsymmetric, positive definite linear systems resulting from the finite element discretization of advection-diffusion equations. A preconditioned GMRES iteration is used to solve a Schur complement system of equations for the subdomain interface variables. In the preconditioning step of each iteration, a partially subassembled interface problem is solved. A convergence rate estimate for the GMRES iteration is established for the cases where the advection is not strong, under the condition that the mesh size is small enough. The estimate deteriorates with a decrease of the viscosity and for fixed viscosity it is independent of the number of subdomains and depends only slightly on the subdomain problem size. Numerical experiments for several two-dimensional advection-diffusion problems illustrate the fast convergence of the proposed algorithm for both diffusion-dominated and advection-dominated cases.
"A balancing domain decomposition method by constraints for advection-diffusion problems." Commun. Appl. Math. Comput. Sci. 3 (1) 25 - 60, 2008. https://doi.org/10.2140/camcos.2008.3.25