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On the Babuška-Osborn approach to finite element analysis: L2 estimates for unstructured meshes
journal contribution
posted on 2023-06-09, 12:18 authored by Charalambos MakridakisCharalambos MakridakisThe 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 MathematikISSN
0029-599XPublisher
Springer VerlagExternal DOI
Issue
4Volume
139Page range
831-844Department affiliated with
- Mathematics Publications
Full text available
- Yes
Peer reviewed?
- Yes
Legacy Posted Date
2018-02-26First Open Access (FOA) Date
2019-02-24First Compliant Deposit (FCD) Date
2018-02-26Usage metrics
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