Lower bounds for three-dimensional nonoverlapping domain decomposition algorithms
The authors show that the known upper bounds of the wire basket preconditioner and the Neumann-Neumann preconditioner in three dimensions for nonoverlapping domain decomposition algorithms are sharp. The proof for the wire basket preconditioner is based on finding that the upper bound of \(\lambda_{\min}\) is of the same order than the known lower bound, and the lower bound of \(\lambda_{\max}\) is \(1\). Then, the condition number for this preconditioner is bounded below and above for the same quantity. The same procedure follows for the Neumann-Neumann preconditioner where in this case \(1\) is an upper bound of \(\lambda_{\min}\), and the lower bound of \(\lambda_{\max}\) is of the same order than the known upper bound.
scientific article; zbMATH DE number 1884039