Algebraic multigrid for discontinuous Galerkin discretizations of heterogeneous elliptic problems

Numerical Linear Algebra with Applications

Volumn 19, Issue 2, 2012, Pages 367-388

  Bastian, Peter ^a   Blatt, Markus ^a   Scheichl, Robert ^b  

^a UNIVERSITY OF HEIDELBERG   (Germany)

^b University of Bath   (United Kingdom)

Author keywords

Aggregation type algebraic multigrid; Elliptic PDEs; Interior penalty discontinuous Galerkin methods; Robustness for heterogeneous coefficients; Subspace correction

Indexed keywords

EID: 84856559857     PISSN: 10705325     EISSN: 10991506     Source Type: Journal    
DOI: 10.1002/nla.1816     Document Type: Article

References (44)

1. Wheeler MF. An elliptic collocation-finite element method with interior penalties. SIAM Journal on Numerical Analysis 1978; 15(1):152-161. DOI: 10.1137/0715010.
2. Arnold DN. An interior penalty finite element method with discontinuous elements. SIAM Journal on Numerical Analysis 1982; 19(4):742-760. DOI: 10.1137/0719052.
3. Oden JT, Babuška I, Baumann CE. A discontinuous hp finite element method for diffusion problems. Journal of Computational Physics 1998; 148(2):491-519. DOI: 10.1006/jcph.1998.6032.
4. Baumann CE, Oden JT. A discontinuous hp finite element method for convection-diffusion problems. Computer Methods in Applied Mechanics and Engineering 1999; 175(3-4):311-341. DOI: 10.1016/S0045-7825(98)00359-4.
5. Rivière BM, Wheeler MF, Girault V. Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems I. Computational Geosciences 1999; 3(3-4):337-360. DOI: 10.1023/A:1011591328604.
6. Arnold DN, Brezzi F, Cockburn B, Marini LD. Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM Journal on Numerical Analysis 2002; 39(5). DOI: 10.1137/S0036142901384162.
7. Rivière B. Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation, Frontiers in Mathematics, Vol.35. SIAM: Philadelphia, PA, USA, 2008.
8. Ern A, Stephansen FA, Zunino P. A discontinuous Galerkin method with weighted averages for advection-diffusion equations with locally small and anisotropic diffusivity. IMA Journal of Numerical Analysis 2009; 29:235-256. DOI: 10.1093/imanum/drm050.
9. Bastian P, Reichenberger V. Multigrid for higher order discontinuous Galerkin finite elements applied to groundwater flow. Technical Report 2000-37, SFB 359 2000.
10. Lasser C, Toselli A. Overlapping preconditioners for discontinuous Galerkin approximations of second order problems. In Domain Decomposition Methods in Science and Engineering. Papers of the Thirteenth International Conference on Domain Decomposition Methods, Lyon, France, October 9-12, 2000, Debit N, Garbay M, Hoppe R, Périaux J, Keyes D, Kuznetsov Y (eds). International Center for Numerical Methods in Engineering (CIMNE): Barcelona, Spain, 2002; 77-84.
11. Feng X, Karakashian OA. Two-level additive Schwarz methods for a discontinuous Galerkin approximation of second order elliptic problems. SIAM Journal on Numerical Analysis 2002; 39(4):1343-1365. DOI: 10.1137/S0036142900378480.
12. Lasser C, Toselli A. An overlapping domain decomposition preconditioner for a class of discontinuous Galerkin approximations of advection-diffusion problems. Mathematics of Computation 2003; 72(243):1215-1238. DOI: 10.1090/S0025-5718-03-01484-4.
13. Hemker PW, Hoffmann W, van Raalte MH. Two-level Fourier analysis of a multigrid approach for discontinuous Galerkin discretizations. SIAM Journal on Scientific Computing 2003; 25(3):1018-1041. DOI: 10.1137/S1064827502405100.
14. Gopalakrishnan J, Kanschat G. A multilevel discontinuous Galerkin method. Numerische Mathematik 2003; 95:527-550.
15. Hemker PW, Hoffmann W, van Raalte MH. Fourier two-level analysis for higher dimensional discontinuous Galerkin discretisation. Computing and Visualization in Science 2004; 7:159-172. DOI: 10.1007/s00791-004-0136-1.
16. Brenner SC, Zhao J. Convergence of multigrid algorithms for interior penalty methods. Applied Numerical Analysis and Computational Mathematics 2005; 2(1):3-18. DOI: 10.1002/anac.200410019.
17. Johannsen K. Multigrid methods for nonsymmetric interior penalty discontinuous Galerkin methods. ICES Report 05-23, University of Texas at Austin, 2005.
18. Dobrev VA, Lazarov RD, Vassilevski PS, Zikatanov LT. Two-level preconditioning of discontinuous Galerkin approximations of second-order elliptic equations. Numerical Linear Algebra with Applications 2006; 13(9):753-770. DOI: 10.1002/nla.504.
19. Dobrev VA. Preconditioning of discontinuous Galerkin methods for second order elliptic problems. PhD Thesis, Texas A&M University, 2007.
20. Antonietti PF. Domain decomposition, spectral correctness and numerical testing of discontinuous Galerkin methods. PhD Thesis, University of Pavia, Italy, 2007.
21. Antonietti PF, Ayuso B. Multiplicative Schwarz methods for discontinuous Galerkin approximations of elliptic problems. ESAIM: Mathematical Modelling and Numerical analysis 2008; 42:443-469. DOI: 10.1051/m2an:2008012.
22. Brix K, Pinto MC, Dahmen W, Massjung R. Multilevel preconditioners for the interior penalty discontinuous Galerkin method II quantitative studies. Communications in Computational Physics 2009; 5(2-4):296-325.
23. Prill F, Lukáčová-Medvid'ová M, Hartmann R. Smoothed aggregation multigrid for the discontinuous Galerkin method. SIAM Journal on Scientific Computing 2009; 31:3503-3528. DOI: 10.1137/080728457.
24. Ayusode Dios B, Zikatanov L. Uniformly convergent iterative methods for discontinuous Galerkin discretizations. Journal of Scientific Computing 2009; 40:4-36. DOI: 10.1007/s10915-009-9293-1.
25. Olson LN, Schroder JB. Smoothed aggregation multigrid solvers for high-order discontinuous Galerkin methods for elliptic problems. Journal of Computational Physics 2011; 230(18):6959-6976. DOI: 10.1016/j.jcp.2011.05.009.
26. Ayuso DeDios B, Holst M, Zhu Y, Zikatanov L. Multilevel preconditioners for discontinuous Galerkin approximations of elliptic problems with jump coefficients. Technical Report. arXiv:1012.1287 2011.
27. Blatt M. A parallel algebraic multigrid method for elliptic problems with highly discontinuous coefficients. PhD Thesis, Ruprecht-Karls-Universität Heidelberg, 2010.
28. Xu J. Iterative methods by space decomposition and subspace correction. SIAM Review 1992; 34(4):581-613. DOI: 10.1137/1034116.
29. Xu J. The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids. Computing 1996; 56(3):215-235. DOI: 10.1007/BF02238513.
30. Hackbusch W. Elliptic Differential Equations. Theory and Numerical Treatment. Springer Verlag: Berlin, 2003.
31. Ern A, Mozolevski I, Schuh L. Discontinuous Galerkin approximation of two-phase flows in heterogeneous porous media with discontinuous capillary pressures. Computer Methods in Applied Mechanics and Engineering 2010; 199(23-24):1491-1501. DOI: 10.1016/j.cma.2009.12.014.
32. Epshteyn Y, Rivière B. Estimation of penalty parameters for symmetric interior penalty Galerkin methods. Journal of Computational and Applied Mathematics 2007; 206:843-872. DOI: 10.1016/j.cam.2006.08.029.
33. Houston P, Hartmann R. An optimal order interior penalty discontinuous Galerkin discretization of the compressible Navier-Stokes equations. Journal of Computational Physics 2008; 227:9670-9685. DOI: 10.1016/j.jcp.2008.07.015.
34. Bastian P. Higher order discontinuous Galerkin methods for flow and transport in porous media. In Challenges in Scientific Computing-CISC 2002, Bänsch E (ed.), No. 35 in Lecture Notes in Computational Science and Engineering. Springer: Berlin Heidelberg, 2003; 1-22.
35. Rivière BM, Wheeler MF, Girault V. A prior error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems. SIAM Journal on Numerical Analysis 2001; 39(3):902-931. DOI: 10.1137/S003614290037174X.
36. Raw M. A coupled algebraic multigrid method for the 3D Navier-Stokes equations. In Fast Solvers for Flow Problems, Proceedings of the 10th GAMM-seminar, Vol.49, Notes on Numerical Fluid Mechanics. Vieweg-Verlag: Braunschweig, Wiesbaden, 1985.
37. Scheichl R, Vainikko E. Additive Schwarz with aggregation-based coarsening for elliptic problems with highly variable coefficients. Computing 2007; 80(4):319-343. DOI: 10.1007/s00607-007-0237-z.
38. Ruge JW, Stüben K. Algebraic multigrid. In Multigrid Methods, McCormick SF (ed.). chap. 4, SIAM: Philadelphia, 1987; 73-130.
39. Vaněk P, Mandel J, Brezina M. Algebraic multigrid based on smoothed aggregation for second and fourth order problems. Computing 1996; 56:179-196.
40. Braess D. Towards algebraic multigrid for elliptic problems of second order. Computing 1995; 55:379-393. DOI: 10.1007/BF02238488.
41. Cliffe KA, Graham IG, Scheichl R, Stals L. Parallel computation of flow in heterogeneous media modelled by mixed finite elements. Journal of Computational Physics 2000; 164:258-282. DOI: 10.1006/jcph.2000.6593.
42. Bastian P, Blatt M, Dedner A, Engwer C, Klöfkorn R, Ohlberger M, Sander O. A generic grid interface for parallel and adaptive scientific computing. Part I: abstract framework. Computing 2008; 82(2-3):103-119. DOI: 10.1007/s00607-008-0003-x.
43. Bastian P, Blatt M, Dedner A, Engwer C, Klöfkorn R, Kornhuber R, Ohlberger M, Sander O. A generic grid interface for parallel and adaptive scientific computing. Part II: implementation and test in DUNE. Computing 2008; 82(2-3):121-138. DOI: 10.1007/s00607-008-0004-9.
44. Blatt M, Bastian P. The iterative solver template library. In Applied Parallel Computing. State of the Art in Scientific Computing, Vol.4699, Kågström B, Elmroth E, Dongarra J, Waśniewski J (eds), Lecture Notes in Computer Science. Springer: Berlin Heidelberg, 2007; 666-675. DOI: 10.1007/978-3-540-75755-9 82.