Inverse Inequalities on Non - Quasiuniform Meshes and Application tothe Mortar Element

We present a range of mesh-dependent inequalities for piecewise constant and continuous piecewise linear nite element functions u deened on locally reened shape-regular (but possibly non-quasiuniform) meshes. These inequalities involve norms of the form kh uk W s;p (() for positive and negative s and , where h is a function which reeects the local mesh diameter in an appropriate way. The only global parameter involved is N, the total number of degrees of freedom in the nite element space, and we avoid estimates involving either the global maximum or minimum mesh diameter. Our inequalities include new variants of inverse inequalities as well as trace and extension theorems. They can be used in several areas of nite element analysis to extend results-previously known only for quasiuniform meshes-to the locally reened case. Here we describe applications to: (i) the theory of nonlinear approximation and (ii) the stability of the mortar element method for locally reened meshes.