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Domain Decomposition Preconditioners for Discontinuous Galerkin Discretizations of Compressible Fluid Flows

Published online by Cambridge University Press:  28 May 2015

Stefano Giani  and
Paul Houston
Stefano Giani*
Affiliation:
School of Engineering and Computing Sciences, Durham University, South Road, Durham, DH1 3LE, UK
Paul Houston*
Affiliation:
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK
*
Corresponding author.Email address:stefano.giani@durham.ac.uk
Corresponding author.Email address:Paul.Houston@nottingham.ac.uk
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Abstract

In this article we consider the application of Schwarz-type domain decomposition preconditioners to the discontinuous Galerkin finite element approximation of the compressible Navier-Stokes equations. To discretize this system of conservation laws, we exploit the (adjoint consistent) symmetric version of the interior penalty discontinuous Galerkin finite element method. To define the necessary coarse-level solver required for the definition of the proposed preconditioner, we exploit ideas from composite finite element methods, which allow for the definition of finite element schemes on general meshes consisting of polygonal (agglomerated) elements. The practical performance of the proposed preconditioner is demonstrated for a series of viscous test cases in both two- and three-dimensions.

Keywords

65F0865N1265N1565N30Composite finite element methodsdiscontinuous Galerkin methodsdomain decompositionSchwarz preconditionerscompressible fluid flows
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 7 , Issue 2 , May 2014 , pp. 123 - 148
Copyright
Copyright © Global Science Press Limited 2014

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References

[1] Amestoy, P. R., Duff, I. S., Koster, J., and L’Excellent, J.-Y.. A fully asynchronous multifrontal solver using distributed dynamic scheduling. SIAM J. Matrix Anal. Appl., 23(1):15–41, 2001.Google Scholar
[2] Amestoy, P. R., Duff, I.S., and L’Excellent, J.-Y.. Multifrontal parallel distributed symmetri-cand unsymmetric solvers. Comput. Methods Appl. Mech. Eng., 184:501–520, 2000.Google Scholar
[3] Amestoy, P. R., Guermouche, A., L’Excellent, J.-Y., and Pralet, S.. Hybrid scheduling for the parallel solution of linear systems. Parallel Computing, 32(2):136–156, 2006.Google Scholar
[4] Antonietti, P.F. and Ayuso, B.. Schwarz domain decomposition preconditioners for discontinuous Galerkin approximations of elliptic problems: non-overlapping case. M2AN Math. Model. Numer. Anal., 41(1):21–54, 2007.Google Scholar
[5] Antonietti, P.F. and Ayuso, B.. Multiplicative Schwarz methods for discontinuous Galerkin approximations of elliptic problems. M2AN Math. Model. Numer. Anal., 42(3):443–469, 2008.Google Scholar
[6] Antonietti, P.F., Giani, S., and Houston, P.. hp–Version composite discontinuous Galerkin methods for elliptic problems on complicated domains. SIAM J. Sci. Comput., 35(3):A1417–A1439, 2013.Google Scholar
[7] Antonietti, P.F., Giani, S., and Houston, P.. Domain decomposition preconditioners for discontinuous Galerkin methods for elliptic problems on complicated domains. J. Sci. Comp., In press.Google Scholar
[8] Antonietti, P.F. and Houston, P.. A class of domain decomposition preconditioners for hp-discontinuous Galerkin finite element methods. J. Sci. Comp., 46(1):124–149, 2011.CrossRefGoogle Scholar
[9] Balay, S., Buschelman, K., Gropp, W.D., Kaushik, D., Knepley, M.G., Mcinnes, L.C., Smith, B.F., and Zhang, H.. PETSc Web page, 2001. http://www.mcs.anl.gov/petsc.Google Scholar
[10] Barker, A.T., Brenner, S.C., Park, E.-H., and Sung, Li-Y.. Two-level additive Schwarz precon-ditioners for a weakly over-penalized symmetric interior penalty method. J. Sci. Comp., 47:27–49, 2011.Google Scholar
[11] Bassi, F., Botti, L., Colombo, A., Di, D.A. Pietro, and Tesini, P.. On the flexibility of agglomeration based physical space discontinuous Galerkin discretizations. J. Comput. Phys., 231(1):45–65, 2012.Google Scholar
[12] Bassi, F. and Rebay, S.. A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations. J. Comp. Phys., 131:267–279, 1997.Google Scholar
[13] Bassi, F. and Rebay, S.. High-order accurate discontinuous finite element solution of the 2d Euler equations. J. Comp. Phys., 138:251–285, 1997.Google Scholar
[14] Baumann, C.E. and Oden, J.T.. A discontinuous hp finite element method for the Euler and Navier-Stokes equations. Internat. J. Numer. Methods Fluids, 31:79–95, 1999.Google Scholar
[15] Baumann, C.E. and Oden, J.T.. An adaptive-order discontinuous Galerkin method for the solution of the Euler equations of gas dynamics. Internat. J. Numer. Methods Engrg., 47:61–73, 2000.Google Scholar
[16] Brenner, S.C. and Wang, K.. Two-level additive Schwarz preconditioners for C0 interior penalty methods. Numer. Math., 102(2):231–255, 2005.Google Scholar
[17] Cangiani, A., Georgoulis, E.H., and Houston, P.. hp-Version discontinuous Galerkin methods on polygonal and polyhedral meshes. Math. Models Methods Appl. Sci., In press.Google Scholar
[18] Dolejší, V.. On the discontinuous Galerkin method for the numerical solution of the Navier-Stokes equations. Int. J. Numer. Meth. Fluids, 45:1083–1106, 2004.Google Scholar
[19] Feistauer, M., Felcman, J., and Stršskraba, I.. Mathematical and Computational Methods for Compressible Flow. Clarendon Press, Oxford, 2003.Google Scholar
[20] Feng, X. and Karakashian, O. A.. Two-level additive Schwarz methods for a discontinuous Galerkin approximation of second order elliptic problems. SIAM J. Numer. Anal., 39(4):1343–1365 (electronic), 2001.Google Scholar
[21] Fidkowski, K.J. and Darmofal, D.L.. A triangular cut-cell adaptive method for high-order discretizations of the compressible Navier-Stokes equations. J. Comput. Phys., 225:1653–1672, 2007.Google Scholar
[22] Fidkowski, K.J., Oliver, T.A., Lu, J., and Darmofal, D.L.. p-Multigrid solution of highorder discontinuous Galerkin discretizations of the compressible Navier-Stokes equations. J. Comput. Phys., 207(1):92–113, July 2005.Google Scholar
[23] Giani, S. and Houston, P.. Anisotropic hp–adaptive discontinuous Galerkin finite element methods for compressible fluid flows. Int. J. Numer. Anal. Model., 9(4):928–949, 2012.Google Scholar
[24] Hackbusch, W. and Sauter, S.A.. Composite finite elements for the approximation of PDEs on domains with complicated micro-structures. Numer. Math., 75:447–472, 1997.Google Scholar
[25] Harriman, K., Houston, P., Senior, B., and Süli, E.. hp–Version discontinuous Galerkin methods with interior penalty for partial differential equations with nonnegative characteristic form. In Shu, C.-W., Tang, T., and Cheng, S.-Y., editors, Recent Advances in Scientific Computing and Partial Differential Equations. Contemporary Mathematics Vol. 330, pages 89–119. AMS, 2003.Google Scholar
[26] Hartmann, R.. The role of the Jacobian in the adaptive discontinuous Galerkin method for the compressible Euler equations. In Warnecke, G., editor, Analysis and Numerics for Conservation Laws, pages 301–316. Springer, 2005.Google Scholar
[27] Hartmann, R. and Houston, P.. Adaptive discontinuous Galerkin finite element methods for the compressible Euler equations. J. Comput. Phys., 183(2):508–532, 2002.Google Scholar
[28] Hartmann, R. and Houston, P.. Symmetric interior penalty DG methods for the compressible Navier–Stokes equations I: Method formulation. Int. J. Num. Anal. Model., 3(1):1–20, 2006.Google Scholar
[29] Hartmann, R. and Houston, P.. Symmetric interior penalty DG methods for the compressible Navier–Stokes equations II: Goal–oriented a posteriori error estimation. Int. J. Num. Anal. Model., 3(2):141–162, 2006.Google Scholar
[30] Hartmann, R. and Houston, P.. An optimal order interior penalty discontinuous Galerkin discretization of the compressible Navier–Stokes equations. J. Comput. Phys., 227(22):9670–9685, 2008.Google Scholar
[31] Hartmann, R. and Leicht, T.. Error estimation and anisotropic mesh refinement for 3d aerodynamic flow simulations. J. Comput. Phys., 229(19), 2010.Google Scholar
[32] Karypis, G. and Kumar, V.. A fast and highly quality multilevel scheme for partitioning irregular graphs. SIAM J. Sci. Comput., 20(1):359–392, 1999.Google Scholar
[33] Oden, J.T. Prudhomme, S., Pascal, F. and Romkes, A.. Review of a priori error estimation for discontinuous Galerkin methods. Technical report, TICAM Report 00–27, Texas Institute for Computational and Applied Mathematics, 2000.Google Scholar
[34] van der Vegt, J.J.W. and van der Ven, H.. Space-time discontinuous Galerkin finite element method with dynamic grid motion for inviscid compressible flows, I. General formulation. J. Comp. Phys., 182:546–585, 2002.Google Scholar
[35] Van der Zee, K.G.. An H1(Ph)-coercive discontinuous Galerkin formulation for the Poisson problem: 1-D Analysis. Master’s thesis, TU Delft, 2004.Google Scholar