Multilevel Schwarz Methods for the Biharmonic Dirichlet Problem

The author considers the solution of the algebraic system of equations that result from the finite element discretization of the biharmonic equation. Some multilevel algorithms are designed and analyzed using a Schwarz framework. Both additive and multiplicative variants of the algorithms are considered and condition number estimates for the additive algorithms and the energy norm estimates for the error propagation operator of the multiplicative algorithms are given. It is noted that for a proper ordering, the iterative operators of the multiplicative algorithms correspond to the error propagation operators of certain V-cycle multigrid methods.