Convergence Analysis of a Multigrid Method for a Convection-Dominated Model Problem

Abstract

The paper presents a convergence analysis of a multigrid solver for a system of linear algebraic equations resulting from the discretization 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 intergrid 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 andthe 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.