Condition number bounds and mesh quality

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.