Estimation and numerical validation of inf-sup constant for bilinear form (p, div u)

We give a derivation for the value of inf-sup constant for the bilinear form (p, div u). We prove that the value of inf-sup constant is equal to 1.0 in all cases and is independent of the size and shape of the domain. Numerical tests for validation of inf-sup constants is performed using finite dimensional spaces defined in \cite{2020jain} on two test domains i) a square of size $\Omega = [0,1]^2$, ii) a square of size $\Omega = [0,2]^2$, for varying mesh sizes and polynomial degrees. The numeric values are in agreement with the theoretical value of inf-sup term.