On the Babuška–Osborn approach to finite element analysis: $L^2$ estimates for unstructured meshes

Abstract

This paper is devoted to a long standing issue in the finite element analysis for elliptic problems. The standard approach to $L^2$ bounds uses the $H^1$ bound in combination to a duality argument, known as Nitsche’s trick, to recover the optimal a priori order of the method. Although this approach makes perfect sense for quasi-uniform meshes, it does not provide the expected information for unstructured meshes since the final estimate involves the maximum mesh size. Babuška and Osborn (Numer Math 34:41–62, 1980), addressed this issue for a one dimensional problem by introducing a technique based on mesh-dependent norms. The key idea was to see the bilinear form posed on two different spaces; equipped with the mesh dependent analogs of $L^2$ and $H^2$ and to show that the finite element space is inf-sup stable with respect to these norms. Although this approach is readily extendable to multidimensional setting, the proof of the inf-sup stability with respect to mesh dependent norms is known only in very limited cases. We establish the validity of the inf-sup condition for standard conforming finite element spaces of any polynomial degree under certain restrictions on the mesh variation which however permit unstructured non quasiuniform meshes. As a consequence we derive $L^2$ estimates for the finite element approximation via quasioptimal bounds and examine related stability properties of the elliptic projection.