|| IGPM223.pdf November 2002 |
|| Convergence Analysis of a Multigrid Solver for a Finite Element Method Applied to Convection-Diffusion Problems |
||Maxim A. Olshanskii, Arnold Reusken|
||The paper presents a convergence analysis of a multigrid solver for a system of
linear algebraic equations resulting from the disretization of a convection-diffusion problem using a finite element method. We consider piecewise linear finite elements in combination with a streamline diffusion stabilization. We analyze a multigrid method that is based on canonical inter-grid transfer operators, a "direct discretization" approach for the coarse-grid operators, a smoother of line-Jacobi
type and one nonstandard component which is called a "local presolver". A robust (diffusion and h-independent) bound for the contraction number of the two-grid method and the multigrid W-cycle are proved for a special class of convection-diffusion problems, namely with Neumann conditions on the out boundary, Dirichlet conditions on the rest of the boundary and a direction that is constant and aligned with gridlines. Our convergence analysis is based on modified smoothing and
approximation properties. The arithmetic complexity of one multigrid iteration is optimal up to a logarithmic term.
|| multigrid, streamline diffusion, convection-diffusion |