Spectral/hp least-squares finite element formulation for the Navier-Stokes equations

  • Authors:

  • J. P. Pontaza;J. N. Reddy

  • Affiliations:

  • Department of Mechanical Engineering, Texas A&M University, College Station, TX;Department of Mechanical Engineering, Texas A&M University, College Station, TX

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2003

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Abstract

We consider the application of least-squares finite element models combined with spectral/hp methods for the numerical solution of viscous flow problems. The paper presents the formulation, validation, and application of a spectral/ hp algorithm to the numerical solution of the Navier-Stokes equations governing two- and three-dimensional stationary incompressible and low-speed compressible flows. The Navier-Stokes equations are expressed as an equivalent set of first-order equations by introducing vorticity or velocity gradients as additional independent variables and the least-squares method is used to develop the finite element model. High-order element expansions are used to construct the discrete model. The discrete model thus obtained is linearized by Newton's method, resulting in a linear system of equations with a symmetric positive definite coefficient matrix that is solved in a fully coupled manner by a preconditioned conjugate gradient method. Spectral convergence of the L2 least-squares functional and L2 error norms is verified using smooth solutions to the two-dimensional stationary Poisson and incompressible Navier-Stokes equations. Numerical results for flow over a backward-facing step, steady flow past a circular cylinder, three-dimensional lid-driven cavity flow, and compressible buoyant flow inside a square enclosure are presented to demonstrate the predictive capability and robustness of the proposed formulation.