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.
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Acknowledgements
The author would like to thank Georgios Akrivis, Manolis Georgoulis, Tristan Pryer and Andreas Veeser for many useful discussions and suggestions regarding this work.
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Partially supported by the EU Horizon 2020 research and innovation programme under the Marie Skodowska-Curie project ModCompShock (modcompshock.eu) Agreement No 642768.
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Makridakis, C.G. On the Babuška–Osborn approach to finite element analysis: \(L^2\) estimates for unstructured meshes. Numer. Math. 139, 831–844 (2018). https://doi.org/10.1007/s00211-018-0955-5
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DOI: https://doi.org/10.1007/s00211-018-0955-5