We investigate the application of a least squares finite element method for the solution of fluid flow problems. The least squares finite element method is based on the minimization of the L2 norm of the equation residuals. Upon discretization, the formulation results in a symmetric, positive definite matrix system which enables efficient iterative solvers to be used. The other motivations behind the development of least squares finite element methods are the applicability of higher order elements and the possibility of using the norm associated to the least squares functional for error estimation. For steady incompressible flows, we develop a method employing linear and quadratic triangular elements and compare their respective accuracy. For steady compressible flows, an implicit conservative least squares scheme which can capture shocks without the addition of artificial viscosity is proposed. A refinement strategy based upon the use of the least squares residuals is developed and several numerical examples are used to illustrate the capabilities of the method when implemented on unstructured triangular meshes.
LEFEBVRE, D., PERAIRE, J. and MORGAN, K. (1992), "LEAST SQUARES FINITE ELEMENT SOLUTION OF COMPRESSIBLE AND INCOMPRESSIBLE FLOWS", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 2 No. 2, pp. 99-113. https://doi.org/10.1108/eb017483
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