Multigrid methods based on matrix-dependent coarse spaces for nonconforming streamline-diffusion finite element discretization of convection-diffusion problems

We apply the streamline-diffusion finite element method for solving convection-diffusion problems. Nonconforming quadrilateral finite elements proposed by Rannacher and Turek are used to discretize the problem. We consider a construction of a pure algebraic multigrid procedure for the solution of the resulting discrete system. The existence of hierarchy of grids is exploited to generate coarse-grid finite element spaces in an algebraic manner, so that the resulting coarse-grid matrices are as sparse as the original stiffness matrix. The proposed multigrid method is used as a preconditioner in the generalized conjugate gradient iterative method for solving the resulting nonsymmetric system of linear algebraic equations. The performance of the method is tested on a two-dimensional convection-diffusion model problem both in the convection-dominated and diffusion-dominated limits.