On the smallest eigenvalue of finite element equations with meshes without regularity assumptions
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A lower bound is provided for the smallest eigenvalue of finite element equations with arbitrary conforming simplicial meshes without any regularity assumptions. The bound has the same form as the one by Graham and McLean [SIAM J. Numer. Anal., 44 (2006), pp. 1487-1513] but doesn't require any mesh regularity assumptions, neither global nor local. This shows that the local mesh regularity condition required by Graham and McLean is not necessary. Equivalently, it can be seen as an improvement of the bound for arbitrary conforming simplicial meshes by Kamenski, Huang, and Xu [Math. Comp., 83 (2014), pp. 2187-2211] from the element-wise form to the patch-wise form. In three and more dimensions, the bound depends on the number of degrees of freedom and the term corresponding to the average mesh non-uniformity. In two dimension, the bound depends on the number of degrees of freedom and a logarithmic term involving number of degrees of freedom and the volume of the smallest patch.
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