University of Sussex
Browse
__smbhome.uscs.susx.ac.uk_bw233_Desktop_SRO_SRO - Charalambos Makridakis_BOInfsup_NumerMath_accepted_final.pdf (205.59 kB)

On the Babuška-Osborn approach to finite element analysis: L2 estimates for unstructured meshes

Download (205.59 kB)
journal contribution
posted on 2023-06-09, 12:18 authored by Charalambos MakridakisCharalambos Makridakis
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.

History

Publication status

  • Published

File Version

  • Accepted version

Journal

Numerische Mathematik

ISSN

0029-599X

Publisher

Springer Verlag

Issue

4

Volume

139

Page range

831-844

Department affiliated with

  • Mathematics Publications

Full text available

  • Yes

Peer reviewed?

  • Yes

Legacy Posted Date

2018-02-26

First Open Access (FOA) Date

2019-02-24

First Compliant Deposit (FCD) Date

2018-02-26

Usage metrics

    University of Sussex (Publications)

    Categories

    No categories selected

    Exports

    RefWorks
    BibTeX
    Ref. manager
    Endnote
    DataCite
    NLM
    DC