Incremental unknowns and graph techniques with in-depth refinement

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Autor:
Garcia, Salvador - Tone, Florentina
URI:
http://repositoriodigital.uct.cl/handle/10925/3786
Datos de publicación:
INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING,Vol.4,143-177,2007
Temas:
finite differences - incremental unknowns - hierarchical basis - Laplace operator - Poisson equation - Chebyshev polynomials - Fejer's kernel
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Resumen:
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(vertical bar logd h vertical bar(3)) for the first order incremental unknowns, and q(d)O(1/H-2)O((logd 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) = (d1)/(2) and q(d) = (d-1)/(2) (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(vertical bar logd(h)vertical bar) 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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