Incremental unknowns and graph techniques with in-depth refinement

Authors
Authors
Date
2012-02-25
Datos de publicación:
International Journal of Numerical Analysis and Modeling, Vol.4, Nº2, 149-177, 2007
Keywords
Matemáticas
Abstract
With in-depth refinement, the condition number of the incremental unknowns matrix associated to the Laplace operator is p(d)O(1/ H2)O( logd<sup/>h 3) for the first order incremental unknowns, and q(d)O(1/H2) O((logdh)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) = d - 1/2 and q(d) = d - 1/2 1/12d(d2 - 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((logdh)2) for the first order incremental unknowns, and q(d)O( logd h ) for the second order incremental unknowns. For comparison, the condition number of the nodal unknowns matrix associated to the Laplace operator is O(1/h2). 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.
Description