Abstract
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.
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 12071350
Funding statement: The authors were partially supported by the Sino-German Center for Research Promotion under Grant GZ 1571. The work of Xuejun Xu was supported by the National Natural Science Foundation of China (Grant No. 12071350).
Acknowledgements
We thank the editor and the anonymous referees, who meticulously read through the paper and made many helpful suggestions which led to an improved presentation of this paper.
References
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