Nested iterations and strengthened Cauchy-Bunyakowski-Schwarz inequalities.

ABSTRACT:
It is known that in one, two, and three spatial dimensions, the optimal constant in the
strengthened Cauchy-Bunyakowski-Schwarz (CBS) inequality for the Laplacian for red-refined linear finite element spaces, takes values zero, $\half\sqrt{2}$ and $\half\sqrt{3}$, respectively. In this paper we will conjecture an explicit relation between these numbers and the spatial dimension, which will also be valid for dimensions four and up. For each individual value of $n$, it is easy to verify the conjecture. Apart from giving additional insight into the matter, the result may find applications in four dimensional finite element codes in the context of computational relativity and financial mathematics.