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 constr aints/variables, the first one aimed at the development of consistent eleme nts and the second one aimed at the development of non-locking and stable e lements. The mentioned tools are rooted in the physics of the continuum mec hanics problem. Mathematicians, on the other side, developed complex and po werful tools to examine the convergence of finite element formulations, suc h as the inf-sup condition, these methods are based on the properties of th e elliptical PDEs that constitute the mathematical model of the continuum m echanics problem. In this paper we intend to understand the inf-sup conditi on from an engineering perspective, so as to be able to incorporate it into the package of tools used in the development of finite element formulation s.