MULTIGRID FORMULATION FOR THE HELMHOLTZ OPERATOR WITH OPEN BOUNDARIES

The application of multigrid methods to the system of equations arising from the finite-element discretization of a two-dimensional waveguide structure has been studied. The problem is described by the scalar Helmholtz equation subjected to Neumann and radiation boundary conditions. The basic principles as well as the difficulties that arise in the standard multigrid method when dealing with the Helmholtz operator are briefly presented. The applicability of a modified version of the multigrid algorithm, namely the nested iteration, is demonstrated by the numerical evaluation of the reflection coefficient for a parallel-plate waveguide. INTRODUCTION Multigrid algorithms have proven to be efficient for the solution of the discretization of elliptic partial differential equations. The basic idea behind these techniques is the use of a sequence of increasingly finer grids by projecting the original problem Au = f onto a family of systems Alul = fl at levels l varying from the finest level lmax (on which the solution u is sought) to a coarsest level lmin. Based on the observation that during the relaxation process of an iterative solver the higher (oscillatory) spatial modes of the error are eliminated quickly while lower (smooth) ones are damped very slowly, the essential multigrid principle is to reduce the oscillatory part with a small number of iterations on the fine grid and then to approximate the smooth part of the error on coarser grids by solving the matrix equation Ae = r (1) where e = u− ū represents the difference between the exact solution u and a certain approximation ū of it and r = f−Aū is the corresponding residual [1]. From an algorithmic point of view, for a V-cycle, after some pre-smoothing steps on the fine level the residual is restricted to the coarse one. Thus we have the right hand side for the residual equation (1) which delivers an approximation of the error by means of an iterative process with e = 0 as an initial guess. This procedure is applied recursively all the way down to the coarsest level where a direct solver is employed to solve the corresponding equation exactly. Starting from this coarsest level then the error estimate on each level is interpolated (prolongated) back to the next finer grid and added to the previous approximate value ū. Additionally, the corrected result on each level serves as the initial value for some post-smoothing iterative steps. At the end of the V-cycle an approximate of the finest-level solution is obtained with a numerical complexity ≈ N log N in total, where N is the order of A. However, in case of indefinite problems it has been shown that standard smoothers (e.g. Jacobi, Gauss-Seidel) become unstable since they amplify the smooth error components [2]. Therefore some different approaches have been proposed for the smoothing process, among them (nonlinear) Krylov subspace methods [2],[4], particularly GMRES [3] as it implies no requirements on the positive-definiteness of the matrix A. In order to avoid the use of expensive nonlinear smoothers we have tried the employ a special version of the multigrid algorithm, known as the nested iteration or Full Multigrid (FMG) (Fig. 1). Rather than on the finest level (as for a V-cycle) we start with the exact solution on the coarsest grid. This is used to obtain an improved (with respect to the smooth error components) initial guess for the multigrid process on the next finer grid. Then we perform a fixed number of V-cycles (one has been proven to be enough). Using a suitable prolongation, which is not necessarily the same as in the V-cycle, we get a new starting iterate for the next finer level. The procedure is recursively repeated until we have reached the finest grid.