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Volume 4, Issue 2
Incremental Unknowns and Graph Techniques with In-Depth Refinement

S. Garcia & F. Tone

Int. J. Numer. Anal. Mod., 4 (2007), pp. 143-177.

Published online: 2007-04

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  • 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.

  • AMS Subject Headings

65N06, 65F35, 65M50

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COPYRIGHT: © Global Science Press

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@Article{IJNAM-4-143, author = {}, title = {Incremental Unknowns and Graph Techniques with In-Depth Refinement}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2007}, volume = {4}, number = {2}, pages = {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.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/857.html} }
TY - JOUR T1 - Incremental Unknowns and Graph Techniques with In-Depth Refinement JO - International Journal of Numerical Analysis and Modeling VL - 2 SP - 143 EP - 177 PY - 2007 DA - 2007/04 SN - 4 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/857.html KW - finite differences, incremental unknowns, hierarchical basis, Laplace operator, Poisson equation, Chebyshev polynomials, Fejér's kernel. AB -

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.

S. Garcia & F. Tone. (1970). Incremental Unknowns and Graph Techniques with In-Depth Refinement. International Journal of Numerical Analysis and Modeling. 4 (2). 143-177. doi:
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