On the numerical modeling of convection-diffusion problems by finite element multigrid preconditioning methods

•Geometric finite element multigrid preconditioning schemes for convection diffusion problems.•New proposed multigrid FE preconditioning methods based on approximate inverses.•Finite element multigrid preconditioning schemes in conjunction with PR2 refinement.•Multigrid preconditioning of Krylov subspace methods with approximate inverses.•Comparative results are presented against classic smoothers.

During the last decades, multigrid methods have been extensively used in order to solve large scale linear systems derived from the discretization of partial differential equations using the finite difference method. The effectiveness of the multigrid method can be also exploited by using the finite element method. Finite Element Approximate Inverses in conjunction with Richardon’s iterative method could be used as smoothers in the multigrid method. Thus, a new class of smoothers based on approximate inverses can be derived. Effectiveness of explicit approximate inverses relies in the fact that they are close approximants to the inverse of the coefficient matrix and are fast to compute in parallel. Furthermore, the proposed class of finite element approximate inverses in conjunction with the explicit preconditioned Richardson method yield improved results against the classic smoothers such as Jacobi method. Moreover, a dynamic relaxation scheme is proposed based on the Dynamic Over/Under Relaxation (DOUR) algorithm. Furthermore, results for multigrid preconditioned Krylov subspace methods, such as GMRES(res), IDR(s) and BiCGSTAB based on approximate inverse smoothing and a dynamic relaxation technique are presented for the steady-state convection-diffusion equation.