\input zb-basic
\input zb-ioport
\iteman{io-port 01151343}
\itemau{Altas, Irfan; Dym, Jonathan; Gupta, Murli M.; Manohar, Ram P.}
\itemti{Multigrid solution of automatically generated high-order discretizations for the biharmonic equation.}
\itemso{SIAM J. Sci. Comput. 19, No.5, 1575-1585 (1998).}
\itemab
The Dirichlet problem for the biharmonic equation is considered. For discretizing this problem finite difference approximations are derived on a nine-point compact stencil using the values of the solution and its gradients as unknowns. These finite difference stencils are obtained by using symbolic software packages. A corresponding Mathematica code for getting a fourth-order approximation scheme is presented. This fourth-order scheme produces more accurate approximations than the classical 13-point stencil or the commonly used splitting into two coupled Poisson equations. For solving the discrete problems several multigrid techniques are discussed. In the experiments presented red-black smoothers with overrelaxation, full weighting for the restriction, and cubic interpolation are used. The numerical examples show that the convergence rate of the multigrid $W$-cycle is independent of the discretization parameter. Furthermore, it is demonstrated that only a few $W$-cycles are necessary within a full multigrid algorithm for getting an approximate solution with an error in the order of the discretization error.
\itemrv{M.Jung (Chemnitz)}
\itemcc{}
\itemut{biharmonic equation; finite difference method; multigrid method; Stokes equation; Dirichlet problem; symbolic software packages; Mathematica code; Poisson equations; red-black smoothers; overrelaxation; numerical examples; convergence}
\itemli{doi:10.1137/S1464827596296970}
\end