231 IGPM231.pdf        June 2003
TITLE Convergence Analysis of a Multigrid Method for a Convection-Dominated Model Problem
AUTHORS Maxim A. Olshanskii, Arnold Reusken
ABSTRACT 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 usingfinite 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 and a smoother of line-Jacobi type. 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 outflow boundary, Dirichlet conditions on the rest of the boundary and a flow 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.
KEYWORDS multigrid, streamline diffusion, convection-diffusion
DOI 10.1137/S0036142902418679
PUBLICATION SIAM journal on numerical analysis 42
1261-1291 (2004)