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In this paper, a class of multilevel preconditioning schemes is presented for solving the linear algebraic systems resulting from the application of Morley nonconforming element approximations to the biharmonic Dirichlet problem. Based on an appropriate space splitting of the finite element spaces associated with the refinements and the abstract Schwarz framework, we prove that the proposed multilevel methods with one smoothing step are optimal, i.e., the convergence rate is independent of the mesh sizes and mesh levels. Moreover, the computational complexity is also optimal since the smoothers are performed only once on each level in the algorithm. Numerical experiments are provided to confirm the optimality of the suggested methods.

An Optimal Multilevel Method with One Smoothing Step for the Morley Element