GRID INDEPENDENT CONVERGENCE OF THE MULTIGRID METHOD

The multigrid method for numerical solution of systems of first-order partial differential equations is analyzed. The smoothing iterations are of polynomial type and are used on all grids. Under certain conditions the convergence of the iterations is independent of the grid size. This is explained by the propagation of smooth error modes out through the boundary and the damping of oscillatory modes. An upper bound is derived on the number of multigrid V-cycles needed to solve a two-dimensional, scalar equation. Numerical experiments with a simple scalar equation and the Euler equations confirm the theoretical results. contrast to the multigrid solution of elliptic equations where the exact solution on the coarsest grid is needed to prove grid independent convergence (7). In this paper an explanation of this difference is offered. We show that under certain conditions the number of multigrid iterations to calculate the solution of a first-order problem with constant coefficients is independent of the grid. The theoretical results are confirmed in numerical experiments using the Euler equations. The numerical discretization of a linear partial differential equation by a finite difference or finite volume method is written

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