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Multilevel Schwarz methods for the biharmonic Dirichlet problem. (English) Zbl 0803.65118
Author’s summary: 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.

##### MSC:
 65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65F10 Iterative numerical methods for linear systems 65F35 Numerical computation of matrix norms, conditioning, scaling 65N15 Error bounds for boundary value problems involving PDEs 31A30 Biharmonic, polyharmonic functions and equations, Poisson’s equation in two dimensions 35J40 Boundary value problems for higher-order elliptic equations
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