A Multigrid Method Based on Cell Orientated Discretization for Convection-diiusion Problems

For the convection-diiusion equation in two dimensions we derive the cell orientated discretization which is based on the method of lines leading to diierential-algebraic equations and their time integration by implicit methods. This approach is well-suited in convection-dominated cases. The eeciency of the method depends on the solution of the arising linear systems mainly. Motivated by the deeniteness properties of these unsymmetric systems we choose a multigrid method with problem adapted transfer operators. The interpretation of the cell orientated discretization in a nite-volume or a Petrov-Galerkin context leads to diierent deenitions of restriction and prolonga-tion. In combination with smoothers which are exact solvers in the convection case (Gauss-Seidel, ILU) we achieve a robust multigrid iteration. Finally we present numerical results concerning the quality of the transfer operators as well as the various smoothing iterations. The independance of the convergence rate from the gridsize and the behaviour of various time integration schemes are also examined.