Application of a minimal compatible element to incompressible and nearly incompressible continuum mechanics

In this note we will explore some applications of the recently constructed piecewise affine, $H^1$-conforming element that fits in a discrete de Rham complex [Christiansen and Hu, Generalized finite element systems for smooth differential forms and Stokes’ problem. Numer. Math. 140 (2018)]. In particular we show how the element leads to locking free methods for incompressible elasticity and viscosity robust methods for the Brinkman model.