Least-squares finite element formulations for viscous incompressible and compressible fluid flows

We present least-squares based finite element formulations for the numerical solution of viscous fluid flows governed by the Navier–Stokes equations, as an alternate approach to the well-known weak form Galerkin finite element formulations. The use of least-squares principles leads to a variational unconstrained minimization problem where compatibility conditions between approximation spaces never arise and the resulting linear algebraic problem will have a symmetric positive definite coefficient matrix. We address the issue of norm equivalence of the least-squares functional and its implications on the resulting finite element model. In particular, we develop an understanding of the compromise that must exist between the optimality and practicality of the finite element model, the latter measured in terms of Ck regularity across inter-element boundaries. We show, through numerical examples, that for the traditional C0 basis such a compromise is possible when high p-levels are used to span the finite element spaces. When a low p-level solution is desired, guidelines are presented to obtain a reliable least-squares collocation solution. Numerical examples are presented to demonstrate the high and low p-level approaches. These include incompressible flow past two circular cylinders in a side-by-side arrangement for gap sizes S/D = 2.0 and 0.85, incompressible flow past a square cylinder, and subsonic, transonic, and supersonic compressible flow past a circular cylinder. In addition, we present a discontinuous least-squares formulation, where Ck regularity across inter-element boundaries is enforced in a weak sense through the least-squares functional—allowing for h- and p-type non-conformities in the computational domain.