What are C and h?: Inequalities for the analysis and design of finite element methods

Increasing mathematical analysis of finite element methods is motivating the inclusion of mesh-dependent terms in new classes of methods for a variety of applications. Several inquualities of functional analysis are often employed in convergence proofs. In the following, Poincaré-Friedrichs inequalities, inverse estimates and least-squares bounds are characterized as tools for the error analysis and practical design of finite element methods with terms that depend on the mesh parameter. Sharp estimates of the constants of these inequalities are provided, and precise definitions of mesh size that arise naturally in the context of different problems in terms of element geometry are derived.