A locally conservative LDG method for the incompressible Navier-Stokes equations |
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Authors: | Bernardo Cockburn Guido Kanschat Dominik Schö tzau. |
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Affiliation: | School of Mathematics, University of Minnesota, Vincent Hall, Minneapolis, Minnesota 55455 ; Institut für Angewandte Mathematik, Universität Heidelberg, Im Neuenheimer Feld~293/294, 69120 Heidelberg, Germany ; Mathematics Department, University of British Columbia, 1984 Mathematics Road, Vancouver, British Columbia V6T 1Z2, Canada |
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Abstract: | In this paper a new local discontinuous Galerkin method for the incompressible stationary Navier-Stokes equations is proposed and analyzed. Four important features render this method unique: its stability, its local conservativity, its high-order accuracy, and the exact satisfaction of the incompressibility constraint. Although the method uses completely discontinuous approximations, a globally divergence-free approximate velocity in is obtained by simple, element-by-element post-processing. Optimal error estimates are proven and an iterative procedure used to compute the approximate solution is shown to converge. This procedure is nothing but a discrete version of the classical fixed point iteration used to obtain existence and uniqueness of solutions to the incompressible Navier-Stokes equations by solving a sequence of Oseen problems. Numerical results are shown which verify the theoretical rates of convergence. They also confirm the independence of the number of fixed point iterations with respect to the discretization parameters. Finally, they show that the method works well for a wide range of Reynolds numbers. |
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Keywords: | Finite element methods discontinuous Galerkin methods incompressible Navier-Stokes equations |
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