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

• Published in: Computer methods in applied mechanics and engineering Vol. 195; no. 19; pp. 2454 - 2494
• Main Authors: Pontaza, J.P., Reddy, J.N.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.04.2006; Elsevier
• Subjects: Compressible flow; Compressible flows; shock and detonation phenomena; Computational techniques; Exact sciences and technology; Fluid dynamics; Fundamental areas of phenomenology (including applications); General theory; Incompressible flow; Least-squares finite element formulations; Mathematical methods in physics; Physics; Spectral/ hp methods; Transonic flows; Incompressible flow; Spectral/ hp methods; Compressible flow; Least-squares finite element formulations; Supersonic flow; Compressible fluid; Transonic flow; Unconstrained optimization; Symmetric tensor; Spectral/hp methods; Galerkin-Petrov method; Finite element method; Weak solution; Symmetric matrix; p type semiconductor; Linear algebra; Variational calculus; Circular cylinder; Modelling; Incompressible fluid; Least square fit; Viscous fluids; Viscous flow; Navier-Stokes equations
• ISSN: 0045-7825; 1879-2138
• DOI: 10.1016/j.cma.2005.05.018

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 C k regularity across inter-element boundaries. We show, through numerical examples, that for the traditional C 0 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 C k 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.