A continuous finite element framework for the pressure Poisson equation allowing non‐Newtonian and compressible flow behavior

Computing pressure fields from given flow velocities is a task arising in various engineering, biomedical and scientific computing applications. The so-called pressure Poisson equation (PPE) derived from the balance of linear momentum provides an attractive framework for such a task. However, the PPE increases the regularity requirements on the pressure and velocity spaces, thereby imposing theoretical and practical challenges for its application. In order to stay within a Lagrangian finite element framework, it is common practice to completely neglect the influence of viscosity and compressibility when computing the pressure, which limits the practical applicability of the pressure Poisson method. In this context, we present a mixed finite element framework which enables the use of this popular technique with generalised Newtonian fluids (e.g., blood) and compressible flows, while allowing standard finite element spaces to be employed for the given data and unknowns. This is accomplished through the use of appropriate vector calculus identities and simple projections of certain flow quantities. In the compressible case, the mixed formulation also includes an additional equation for retrieving the density field from the given velocities so that the pressure can be accurately determined. The potential of this new approach is showcased through numerical examples.