Makridakis, Charalambos G (2018) On the Babuška-Osborn approach to finite element analysis: L2 estimates for unstructured meshes. Numerische Mathematik, 139 (4). pp. 831-844. ISSN 0029-599X

PDF - Accepted Version Restricted to SRO admin only until 24 February 2019. Download (210kB)

Abstract

The standard approach to L2 bounds uses theH1 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, [1], 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 L2 and H2 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 L2 estimates for the finite element approximation via quasioptimal bounds and examine related stability properties of the elliptic projection.

Item Type:	Article
Schools and Departments:	School of Mathematical and Physical Sciences > Mathematics
Subjects:	Q Science > QA Mathematics
Depositing User:	Billy Wichaidit
Date Deposited:	26 Feb 2018 09:49
Last Modified:	03 Sep 2018 08:20
URI:	http://srodev.sussex.ac.uk/id/eprint/74084

View download statistics for this item

📧 Request an update