A New Semialgebraic Two-Grid Method for Oseen Problems

Abstract

We consider the numerical solution of discrete Oseen problems. We propose a new approach that consists of first applying a simple algebraic transformation to the linear system, which is afterwards preconditioned with an aggregation-based algebraic two-grid method. An algebraic analysis is provided, which proves uniform convergence in norm with respect to problem parameters if a few constants can be uniformly bounded. A further analysis of these constants shows that they can be bounded in the case of a constant convection field, provided that the coarsening of the pressure unknowns is also driven by the convection field. This makes the method essentially different from a similar method developed for Stokes equations which initially inspired the present work. Technically, this means that one has to either use point-based coarsening, or that an auxiliary convection-diffusion matrix has to be built on the pressure space to guide the coarsening, which makes the method only semialgebraic. Using this ingredient, promising results are obtained, showing that the number of iterations is, in practice, uniformly bounded with respect to both the mesh size and the Reynolds number even for challenging recirculating convection fields or in presence of outflow boundary conditions.