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Consider a domain $\Omega \subseteq \mathbb{R}^{2,3}$ which is non convex and with $C^2$ boundary. Could you recommend a good reference where it is explained how:

  • without needing isoparametric elements
  • with a triangulation of simplices

one can obtain:

  • some space of linear finite elements $S^h_0$ which vanish on $\partial \Omega$ (the reference should clearly explain how to define the hat functions outside of $\Omega$ and what properties the triangulation should satisfy...)
  • a suitable definition of the nodal interpolator $I_h:H^2(\Omega)\cap H^1_0(\Omega)\rightarrow S^h_0$

such that the approximation property $||I_hv-v||_{L^2}+h||\nabla(I_hv-v)||_{L^2}\leq C h^s , ||v||_{H^s}$ holds for $s=1,2$?

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  • $\begingroup$ How is ${\mathbb R}^{2,3}$ defined? $\endgroup$
    – Wolfgang Bangerth
    May 9, 2022 at 15:26
  • $\begingroup$ It is a compact notation to indicate either R² or R³ (@WolfgangBangerth) $\endgroup$
    – Lilla
    May 10, 2022 at 16:16

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