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Condition number bounds and mesh quality
Author(s) -
Branets L.,
Carey G. F.
Publication year - 2010
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.758
Subject(s) - jacobian matrix and determinant , mathematics , condition number , hexahedron , finite element method , quadrilateral , upper and lower bounds , polygon mesh , distortion (music) , stiffness matrix , mesh generation , hessian matrix , matrix (chemical analysis) , algorithm , computer science , eigenvalues and eigenvectors , geometry , mathematical analysis , amplifier , computer network , physics , materials science , bandwidth (computing) , quantum mechanics , composite material , thermodynamics
We analyze the effect of mesh distortion on the condition number of the representative mass matrix M and stiffness matrix K arising in a typical finite element scheme. Bounds are stated for the respective condition numbers in terms of the Jacobian of the map from a reference element. These results are then used to construct the related bounds in terms of representative metrics for mesh distortion. These bounds are easily pre‐computable and provide a new explicit mathematical relation between matrix conditioning and mesh quality metrics. Numerical studies for a 2D test problem using a representative cell quality metric demonstrate the upper bound property and the dependence on cell quality for a quadrilateral cell. Analogous results for a 3D test problem under progressive symmetric mesh distortion of an interior hexahedral cell are also provided, as well as a study on a complex 3D geometry. We conclude by presenting practical adaptive mesh grading applications employing aforementioned mesh quality metrics. Copyright © 2010 John Wiley & Sons, Ltd.