Inverse Estimates for non-quasiuniform meshes and applications

Inverse estimates are typically needed for the analysis of finite element and boundary element methods when it is required to have estimates for a higher norm of some quantity (in an interpolation scale) in terms of some lower norm.

In the finite element case the proofs of inverse estimates have classically required rather strong assumptions of mesh regularity.

Recently we have been developing more flexible methods of proof  which cover first of all the shape regular (non-quasiuniform) case and also the degenerate (non-shape-regular case). We have applied our results to the analysis of mortar (finite) element methods and to quadrature and panel-clustering for boundary elements on stretched meshes.


There are two papers:

W. Dahmen, B. Faermann, I.G. Graham, W. Hackbusch, S.A. Sauter, Inverse Inequalities on Non-Quasiuniform Meshes and Application to the Mortar Element Method, Preprint 24, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig,  Mathematics of Computation,  73  (2004),1107-1138. 

I.G. Graham, W. Hackbusch, S.A. Sauter, Finite Elements on Degenerate Meshes: Inverse-type Inequalities and Applications , Preprint 102(2002), Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig,  to appear in  IMA J. Numer. Anal.. Preprint, various formats