Super-convergence of Discontinuous Galerkin Method Applied to the Navier-Stokes Equations

The practical benefits of the hyper-accuracy properties of the discontinuous Galerkin method are examined. In particular, we demonstrate that some flow attributes exhibit super-convergence even in the absence of any post-processing technique. Theoretical analysis suggest that flow features that are dominated by global propagation speeds and decay or growth rates should be super-convergent. Several discrete forms of the discontinuous Galerkin method are applied to the simulation of unsteady viscous flow over a two-dimensional cylinder. Convergence of the period of the naturally occurring oscillation is examined and shown to converge at 2p+1, where p is the polynomial degree of the discontinuous Galerkin basis. Comparisons are made between the different discretizations and with theoretical analysis.

[1]  J. Douglas,et al.  Interior Penalty Procedures for Elliptic and Parabolic Galerkin Methods , 1976 .

[2]  Jennifer K. Ryan,et al.  Extension of a Post Processing Technique for the Discontinuous Galerkin Method for Hyperbolic Equations with Application to an Aeroacoustic Problem , 2005, SIAM J. Sci. Comput..

[3]  Mark Ainsworth,et al.  Dispersive and dissipative behaviour of high order discontinuous Galerkin finite element methods , 2004 .

[4]  Guido Kanschat,et al.  The local discontinuous Galerkin method for the Oseen equations , 2003, Math. Comput..

[5]  Chi-Wang Shu,et al.  The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems , 1998 .

[6]  M. D. Salas,et al.  On Problems Associated with Grid Convergence of Functionals , 2009 .

[7]  H. L. Atkins.  Local Analysis of Shock Capturing Using Discontinuous Galerkin Methodology , 1997 .

[8]  Karen Dragon Devine,et al.  A posteriori error estimation for discontinuous Galerkin solutions of hyperbolic problems , 2002 .

[9]  Endre Süli,et al.  Post-Processing of Galerkin Methods for Hyperbolic Problems , 2000 .

[10]  Harold L. Atkins,et al.  Two-dimensional Wave Analysis of the Discontinuous Galerkin Method with Non-Uniform Grids and Boundary Conditions , 2002 .

[11]  K. W. Morton,et al.  Finite element methods for hyperbolic equations , 1987 .

[12]  Robert B. Lowrie,et al.  Compact higher-order numerical methods for hyperbolic conservation laws. , 1996 .

[13]  Juhani Pitkäranta,et al.  An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation , 1986 .

[14]  Carl De Boor,et al.  Mathematical aspects of finite elements in partial differential equations : proceedings of a symposium conducted by the Mathematics Research Center, the University of Wisconsin--Madison, April 1-3, 1974 , 1974 .

[15]  H. Atkins,et al.  Quadrature-Free Implementation of the Discontinuous Galerkin Method for Hyperbolic Equations , 1996 .

[16]  Endre Süli,et al.  Enhanced accuracy by post-processing for finite element methods for hyperbolic equations , 2003, Math. Comput..

[17]  P. Raviart,et al.  On a Finite Element Method for Solving the Neutron Transport Equation , 1974 .

[18]  D. L. Young,et al.  A novel immersed boundary procedure for flow and heat simulations with moving boundary , 2009 .

[19]  Harold L. Atkins,et al.  QUADRATURE-FREE IMPLEMENTATION OF DISCONTINUOUS GALERKIN METHOD FOR HYPERBOLIC EQUATIONS , 1996 .

[20]  Jennifer K. Ryan,et al.  Application of an Accuracy Enhancing Post-Processing Method to Aeroacoustic Simulations , 2003 .

[21]  Chi-Wang Shu,et al.  Analysis of the discontinuous Galerkin method applied to the diffusion operator , 1999 .

[22]  B. T. Helenbrook,et al.  Application of “ p ”-multigrid to discontinuous Galerkin formulations of the Poisson equation , 2008 .

[23]  Joseph E. Flaherty,et al.  High-Order Finite Element Methods for Singularly Perturbed Elliptic and Parabolic Problems , 1995, SIAM J. Appl. Math..

[24]  G. Richter.  An Optimal-Order Error Estimate for the Discontinuous Galerkin Method , 1988 .

[25]  J. Z. Zhu,et al.  The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique , 1992 .