@Article{IJNAM-4-2,
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 = {<p style="text-align: justify;">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.</p>},
issn = {2617-8710},
doi = {https://doi.org/2007-IJNAM-857},
url = {https://global-sci.com/article/83764/incremental-unknowns-and-graph-techniques-with-in-depth-refinement}
}