# Reference request for finite elements theory

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 $$s=1,2$$?

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