TY - JOUR
T1 - Optimal multilevel methods for graded bisection grids
AU - Chen, Long
AU - Nochetto, Ricardo H.
AU - Xu, Jinchao
N1 - Funding Information:
L. Chen was supported in part by NSF Grant DMS-0505454, DMS-0811272, and in part by 2010–2011 UC Irvine Academic Senate Council on Research, Computing and Libraries (CORCL). R. H. Nochetto was supported in part by NSF Grant DMS-0505454 and DMS-0807811. J. Xu was supported in part by NSF DMS-0609727, DMS 0915153, NSFC-10528102 and Alexander von Humboldt Research Award for Senior US Scientists.
PY - 2012/1
Y1 - 2012/1
N2 - We design and analyze optimal additive and multiplicative multilevel methods for solving H 1 problems on graded grids obtained by bisection. We deal with economical local smoothers: after a global smoothing in the finest mesh, local smoothing for each added node during the refinement needs to be performed only for three vertices - the new vertex and its two parent vertices. We show that our methods lead to optimal complexity for any dimensions and polynomial degree. The theory hinges on a new decomposition of bisection grids in any dimension, which is of independent interest and yields a corresponding decomposition of spaces. We use the latter to bridge the gap between graded and quasi-uniform grids, for which the multilevel theory is well-established.
AB - We design and analyze optimal additive and multiplicative multilevel methods for solving H 1 problems on graded grids obtained by bisection. We deal with economical local smoothers: after a global smoothing in the finest mesh, local smoothing for each added node during the refinement needs to be performed only for three vertices - the new vertex and its two parent vertices. We show that our methods lead to optimal complexity for any dimensions and polynomial degree. The theory hinges on a new decomposition of bisection grids in any dimension, which is of independent interest and yields a corresponding decomposition of spaces. We use the latter to bridge the gap between graded and quasi-uniform grids, for which the multilevel theory is well-established.
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U2 - 10.1007/s00211-011-0401-4
DO - 10.1007/s00211-011-0401-4
M3 - Article
AN - SCOPUS:84855252687
SN - 0029-599X
VL - 120
SP - 1
EP - 34
JO - Numerische Mathematik
JF - Numerische Mathematik
IS - 1-6
ER -