Efficient modeling of three-dimensional convection-diffusion problems in stationary flows

A large number of transport processes in physics, chemistry, and engineering are described by a convection-diffusion equation. This equation is notoriously difficult to solve due to the presence of convection-related first-order gradient differential operators. We describe a new and efficient numerical method for solving the convection-diffusion equation for laminar flows within channels of arbitrary cross section. It is based on reducing the convection-diffusion equation to a set of pure diffusion equations with a complex-valued potential for which fast and numerically stable solvers are readily available. Additionally, we use an eigenvector projection method that allows us to compute snapshots of full concentration distributions over millions of finite elements within a few seconds using a conventional state-of-the-art desktop computer. Our results will be important for all applications where diffusion and convection are both important for correctly describing material transport.