On the convergence of incompressible finite element formulations - The Patch Test and the inf-sup condition

Engineers have developed robust and efficient incompressible finite element formulations using tools such as the Patch Test and the counting of constraints/variables, the first one aimed at the development of consistent elements and the second one aimed at the development of non-locking and stable elements. The mentioned tools are rooted in the physics of the continuum mechanics problem. Mathematicians, on the other side, developed complex and powerful tools to examine the convergence of finite element formulations, such as the inf-sup condition, these methods are based on the properties of the elliptical PDEs that constitute the mathematical model of the continuum mechanics problem. In this paper we intend to understand the inf-sup condition from an engineering perspective, so as to be able to incorporate it into the package of tools used in the development of finite element formulations.