Year: 2007
International Journal of Numerical Analysis and Modeling, Vol. 4 (2007), Iss. 2 : pp. 143–177
Abstract
With in-depth refinement, the condition number of the incremental unknowns matrix associated to the Laplace operator is $p(d)O(1/H^2)O(|log_d h|^3)$ for the first order incremental unknowns, and $q(d)O(1/H^2)O((log_d h)^2)$ for the second order incremental unknowns, where $d$ is the depth of the refinement, $H$ is the mesh size of the coarsest grid, $h$ is the mesh size of the finest grid, $p(d) = \frac{d-1}{2}$ and $q(d) = \frac{d-1}{2} \frac{1}{12}d(d^2-1)$. Furthermore, if block diagonal (scaling) preconditioning is used, the condition number of the preconditioned incremental unknowns matrix associated to the Laplace operator is $p(d) O((log_d h)^2)$ for the first order incremental unknowns, and $q(d)O(|log_dh|)$ for the second order incremental unknowns. For comparison, the condition number of the nodal unknowns matrix associated to the Laplace operator is $O(1/h^2)$. Therefore, the incremental unknowns preconditioner is efficient with in-depth refinement, but its efficiency deteriorates at some rate as the depth of the refinement grows.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/2007-IJNAM-857
International Journal of Numerical Analysis and Modeling, Vol. 4 (2007), Iss. 2 : pp. 143–177
Published online: 2007-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 35
Keywords: finite differences incremental unknowns hierarchical basis Laplace operator Poisson equation Chebyshev polynomials Fejér's kernel.