Bounds on the Smallest Eigenvalue of Finite Element Equations with Adaptive (anisotropic) Meshes without Regularity Assumptions

Abstract

Highly adapted or misshaped meshes can have a strong influence on the conditioning of the finite element (FE) equations, which, in turn, has an impact on convergence properties and accuracy of iterative methods for solving the resulting linear systems. In most of the available work, some local or global mesh regularity properties are required in order to bound the constants during the derivation of the estimates, which causes degeneration of these estimates in case of extreme mesh geometries or strong irregular anisotropic adaptation. The provided bound for the smallest eigenvalue of the FE equations has a similar 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. In particular, it is valid for highly adaptive, anisotropic, or non-regular meshes without any restrictions.