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Algebraic multigrid techniques for discontinuous Galerkin methods with varying polynomial order

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Abstract

We present a parallel algebraic multigrid (AMG) algorithm for the implicit solution of the Darcy problem discretized by the discontinuous Galerkin (DG) method that scales optimally for regular and irregular meshes. The main idea centers on recasting the preconditioning problem so that existing AMG solvers for nodal lower order finite elements can be leveraged. This is accomplished by a transformation operator which maps the solution from a Lagrange basis representation to a Legendre basis representation. While this mapping function must be user supplied, we demonstrate how easily it can be constructed for somepopular finite element representations includingquadrilateral/hexahedral and triangular/tetrahedral DG formulations. Furthermore, we show that the mapping does not depend on the Jacobian transformation between reference and physical space and so it can be constructed with very limited mesh information. Parallel performance studies demonstrate the versatility of this approach.

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References

  1. Antonietti, P., Ayuso, B.: Two-level Schwarz preconditioners for super penalty discontinuous Galerkin methods. Commun. Comput. Phys. 5, 398–412 (2009)

    Google Scholar 

  2. Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39, 1749–1779 (2002)

    Article  Google Scholar 

  3. Atkins, H., Helenbrook, B.: Application of p-multigrid to discontinuous Galerkin formulations of the Poisson operator. In: AIAA Computational Fluid Dynamics Conference (2005)

  4. Ayuso De Dios, B., Zikatanov, L.: Uniformly convergent iterative methods for discontinuous Galerkin discretizations. J. Sci. Comput. 40, 4–36 (2009)

    Article  Google Scholar 

  5. Barker, A., Brenner, S., Sung, L.: Overlapping Schwarz domain decomposition preconditioners for the local discontinuous Galerkin method for elliptic problems. J. Numer. Math. Appl. 19, 165–187 (2011)

    Google Scholar 

  6. Bastian, P., Blatt, M., Scheichl, R.: Algebraic multigrid for discontinuous galerkin discretizations of heterogeneous elliptic problems. Numer. Linear Algebra Appl. 19, 367–388 (2012)

    Article  Google Scholar 

  7. Beck, R.: Algebraic multigrid by component splitting for edge elements on simplicial triangulations. Tech. Rep. SC 99-40, Zuse Institute, Berlin (1999)

  8. Blackford, L., Demmel, J., Dongarra, J., Duff, I., Hammarling, S., Henry, G., Heroux, M., Kaufman, L., Lumsdaine, A., Petitet, A., Pozo, R., Remington, K., Whaley, R.: An updated set of basic linear algebra subprograms (BLAS). ACM Trans. Math. Soft. 28, 135–151 (2002)

    Article  Google Scholar 

  9. Bochev, P.B., Hu, J.J., Siefert, C.M., Tuminaro, R.S.: An algebraic multigrid approach based on a compatible gauge reformulation of Maxwell’s equations. SIAM J. Sci. Comput. 31, 557–583 (2008)

    Article  Google Scholar 

  10. Brandt, A., McCormick, S.F., Ruge, J.W.: Algebraic multigrid (AMG) for sparse matrix equations. In: Evans, D.J. (ed.) Sparsity and Its Applications, pp 257–284. Cambridge University Press, Cambridge (1984)

    Google Scholar 

  11. Brenner, S., Zhao, J.: Convergence of multigrid algorithms for interior penalty methods. Appl. Numer. Anal. Comput. Math. 2, 3–18 (2005)

    Article  Google Scholar 

  12. Briggs, W.L., Henson, V.E., McCormick, S.: A multigrid tutorial, 2nd edn. SIAM, Philadelphia (2000)

    Book  Google Scholar 

  13. Clees, T.: AMG strategies for PDE systems with applications in industrial semiconductor simulation. PhD thesis, University of Koln (2005)

  14. C. Development Team: CUBIT geometry and meshing toolkit. Version 13.2 (2012)

  15. Dobrev, V.A., Lazarov, R.D., Vassilevski, P.S., Zikatanov, L.T.: Two-level preconditioning of discontinuous Galerkin approximations of second-order elliptic equations. Numer. Linear Algebra Appl. 13, 753–770 (2006)

    Article  Google Scholar 

  16. Dubiner, M.: Spectral methods on triangles and other domains. J. Sci. Comput. 6, 345–390 (1991)

    Article  Google Scholar 

  17. Fidkowski, K.J., Oliver, T.A., Lu, J., Darmofal, D.L.: p-multigrid solution of high-order discontinuous Galerkin discretizations of the compressible Navier-Stokes equations. J. Comput. Phys. 207, 92–113 (2005)

    Article  Google Scholar 

  18. Füllenbach, T, Stüben, K.: Algebraic multigrid for selected PDE systems. In: Proceedings of the Fourth European Conference on Elliptic and Parabolic Problems, pp. 399–410 (2002)

  19. Gee, M., Hu, J., Sala, M., Siefert, C., Tuminaro, R.: ML 5.0 smoothed aggregation user’s guide. Tech. Rep. SAND2006-2649, Sandia National Laboratories (2007)

  20. Gee, M., Siefert, C., Hu, J., Tuminaro, R., Sala, M.: ML 5.0 smoothed aggregation user’s guide, Tech. Rep. SAND2006-2649, Sandia National Laboratories (2006)

  21. Gerstenberger, A., Scovazzi, G., Collis, S.S.: Computing gravity-driven viscous fingering in complex subsurface geometries: a high-order discontinuos Galerkin approach, Computational Geosciences, (submitted)

  22. Gopalakrishnan, J., Kanschat, G.: A multilevel discontinuous Galerkin method. Numer. Math. 95, 527–550 (2003)

    Article  Google Scholar 

  23. Hackbusch, W.: Multigrid Methods and Applications, Vol. 4 of Computational Mathematics. Springer, Berlin (1985)

    Google Scholar 

  24. Hageman, L., Young, D.: Applied Iterative Methods. Academic Press, New York (1981)

    Google Scholar 

  25. Hemker, P.W., Hoffmann, W., Raalte, M.H.v.: Two-level Fourier analysis of a multigrid approach for discontinuous Galerkin discretization. SIAM J. Sci. Comput. 25, 1018–1041 (2003)

    Article  Google Scholar 

  26. Henson, V. E., Yang, U.M.: BoomerAMG: a parallel algebraic multigrid solver and preconditioner. Appl. Numer. Math. Trans. IMACS 41, 155–177 (2002)

    Article  Google Scholar 

  27. Heroux, M., Bartlett, R., Hoekstra, V.H.R., Hu, J., Kolda, T., Lehoucq, R., Long, K., Pawlowski, R., Phipps, E., Salinger, A., Thornquist, H., Tuminaro, R., Willenbring, J., Williams, A.: An overview of Trilinos. Tech. Rep. SAND2003-2927, Sandia National Laboratories (2003)

  28. Hiptmair, R., Xu, J.: Nodal auxiliary space preconditioning in H(curl) and H(div) spaces. SIAM J. Numer. Anal. 45, 2483–2509 (2007)

    Article  Google Scholar 

  29. Jones, J. E., Vassilevski, P.S.: Amge based on element agglomeration. SIAM J. Sci. Comput. 23, 109–133 (2001)

    Article  Google Scholar 

  30. Kanschat, G.: Preconditioning methods for local discontinuous Galerkin discretizations. SIAM J. Sci. Comput. 25, 815–831 (2003)

    Article  Google Scholar 

  31. Kanschat, G.: Robust smoothers for high-order discontinuous Galerkin discretizations of advection-diffusion problems. J. Comput. Appl. Math. 218, 53–60 (2008). Special Issue: Finite Element Methods in Engineering and Science (FEMTEC2006)

    Article  Google Scholar 

  32. Kolev, T., Pasciak, J., Vassilevski, P.: H(curl) auxiliary mesh preconditioning. Tech. Rep. UCRL-JRNL-224227, Lawrence Livermore National Labs (2006)

  33. Kolev, T., Vassilevski, P.: Parallel H1-based auxiliary space AMG solver for H(curl) problems. Tech. Rep. UCRL-TR-222763, Lawrence Livermore National Labs (2006)

  34. Kolev, T., Vassilevski, P.: Some experience with a H1-based auxiliary space AMG for H(curl) problems. Tech. Rep. UCRL-TR-21841, Lawrence Livermore National Labs (2006)

  35. Nastase, C., Mavriplis, D.: A parallel hp-multigrid solver for three-dimensional discontinuous Galerkin discretizations of the Euler equations. AIAA Paper 512 (2007)

  36. L. Olson: Algebraic multigrid preconditioning of high-order spectral elements for elliptic problems on a simplicial mesh. SIAM J. Sci. Comput. 29, 2189–2209 (electronic) (2007)

    Article  Google Scholar 

  37. Olson, L.N., Schroder, J.B.: Smoothed aggregation multigrid solvers for high-order discontinuous galerkin methods for elliptic problems. J. Comput. Phys. 230, 6959–6976 (2011)

    Article  Google Scholar 

  38. Prill, F., Lukáčová-Medvidóvá, M., Hartmann, R.: Smoothed aggregation multigrid for the discontinuous Galerkin method. SIAM J. Sci. Comput. 31, 3503–3528 (2009)

    Article  Google Scholar 

  39. Prometheus. http://www.columbia.edu/ma2325/prometheus/

  40. Tuminaro, J. X. R. S., Zhu, Y.: Auxiliary space preconditioners for mixed finite element methods. In: Widlund, O., Keyes, D.E. (eds.) Domain Decomposition Methods in Science and Engineering XVIII. Vol. 70 of Lecture Notes in Computational Science and Engineering, pp 99–109. Springer, Berlin (2009)

    Chapter  Google Scholar 

  41. Ruge, J.W., Stüben, K.: Algebraic multigrid (AMG). In: McCormick, S.F. (ed.) Multigrid Methods. Frontiers Appl. Math., pp 73–130. SIAM, Philadelphia (1987)

    Chapter  Google Scholar 

  42. Ruge, J.W., Stüben, K.: Algebraic multigrid (AMG). In: McCormick, S.F. (ed.) Multigrid Methods. Vol. 3 of Frontiers in Applied Mathematics, pp 73–130. SIAM, Philadelphia (1987)

    Google Scholar 

  43. Scovazzi, G., Gerstenberger, A., Collis, S.S.: A discontinuous Galerkin method for gravity-driven viscous fingering instabilities in porous media. J. Comput. Phys. (2012)

  44. Shahbazi, K., Mavriplis, D. J., Burgess, N.K.: Multigrid algorithms for high-order discontinuous Galerkin discretizations of the compressible navier-stokes equations. J. Comput. Phys. 228, 7917–7940 (2009)

    Article  Google Scholar 

  45. Sherwin, S., Karniadakis, G.: A new triangular and tetrahedral basis for high-order (hp) finite element methods. Int. J. Numer. Methods Eng. 38, 3775–3802 (1995)

    Article  Google Scholar 

  46. Trottenberg, U., Oosterlee, C., Schüller, A.: Multigrid. Academic, London (2001)

    Google Scholar 

  47. Vaněk, P., Brezina, M., Mandel, J.: Convergence of algebraic multigrid based on smoothed aggregation. Numer. Math. 88, 559–579 (2001)

    Article  Google Scholar 

  48. Vaněk, P., Mandel, J., Brezina, M.: Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems. Computing 56, 179–196 (1996)

    Article  Google Scholar 

  49. Xu, J.: The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids. Computing 56, 215–235 (1996)

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. Computational Shock and Multiphysics Department, Sandia National Laboratories, P.O. Box 5800, MS 1320, Albuquerque, NM, 87185-1320, USA

    C. Siefert

  2. Numerical Analysis and Applications Department, Sandia National Laboratories, P.O. Box 5800, MS 1319, Albuquerque, NM, 87185-1319, USA

    R. Tuminaro, A. Gerstenberger & S. S. Collis

  3. Department of Civil and Environmental Engineering, Duke University, Room 121 Hudson Hall, Box 90287, Durham, NC, 27708-0287, USA

    G. Scovazzi

Authors
  1. C. Siefert
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  2. R. Tuminaro
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  3. A. Gerstenberger
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  4. G. Scovazzi
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  5. S. S. Collis
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Correspondence to C. Siefert.

Additional information

Sandia National Laboratories is a multiprogram laboratory managed and operated, by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the US Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000. The authors would like to acknowledge the support of Department of Energy’s Office of Science through the SciDAC-e Research Grant “Algebraic Multi-Grid Methods for Modeling and Simulation of Carbon Sequestration Processes on Multi-Core/GPU Architectures,” No. 10-014677. SAND Number 2012-9105J.

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Siefert, C., Tuminaro, R., Gerstenberger, A. et al. Algebraic multigrid techniques for discontinuous Galerkin methods with varying polynomial order. Comput Geosci 18, 597–612 (2014). https://doi.org/10.1007/s10596-014-9419-x

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  • DOI: https://doi.org/10.1007/s10596-014-9419-x

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