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 - <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>