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

TL;DR

This paper derives and confirms that the inf-sup constant for the bilinear form (p, div u) is always 1, regardless of domain shape or size, through theoretical proof and numerical validation.

Contribution

The paper provides a theoretical proof that the inf-sup constant for (p, div u) is always 1, and validates this with numerical tests across different domains and discretizations.

Findings

Inf-sup constant equals 1 in all tested cases.

Numerical results agree with theoretical prediction.

Constant is independent of domain size and shape.

Abstract

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.