EDP Sciences logo
Subscriber Authentication Point
Search
Menu
All issues Volume 50 / No 3 (May-June 2016) ESAIM: M2AN, 50 3 (2016) 699-725 References
Issue
ESAIM: M2AN
Volume 50, Number 3, May-June 2016
Special Issue – Polyhedral discretization for PDE
Page(s) 699 - 725
DOI https://doi.org/10.1051/m2an/2015059
Published online 23 May 2016
  1. P.F. Antonietti and B. Ayuso, Schwarz domain decomposition preconditioners for discontinuous Galerkin approximations of elliptic problems: non-overlapping case. M2AN 41 (2007) 21–54. [Google Scholar]
  2. P.F. Antonietti and B. Ayuso, Multiplicative Schwarz methods for discontinuous Galerkin approximations of elliptic problems. M2AN 42 (2008) 443–469. [CrossRef] [EDP Sciences] [MathSciNet] [Google Scholar]
  3. P.F. Antonietti and P. Houston, A class of domain decomposition preconditioners for hp-discontinuous Galerkin finite element methods. J. Sci. Comput. 46 (2011) 124–149. [CrossRef] [MathSciNet] [Google Scholar]
  4. P.F. Antonietti, S. Giani and P. Houston, hp-Version composite discontinuous Galerkin methods for elliptic problems on complicated domains. SIAM J. Sci. Comput. 35 (2013) A1417–A1439. [CrossRef] [Google Scholar]
  5. P.F. Antonietti, S. Giani and P. Houston, Domain decomposition preconditioners for Discontinuous Galerkin methods for elliptic problems on complicated domains. J. Sci. Comput. 60 (2014) 203–227. [CrossRef] [MathSciNet] [Google Scholar]
  6. P.F. Antonietti, P. Houston, M. Sarti and M. Verani, Multigrid algorithms for hp-version interior penalty discontinuous Galerkin methods on polygonal and polyhedral meshes. Preprint arXiv:1412.0913 (2014). [Google Scholar]
  7. P.F. Antonietti, M. Sarti and M. Verani, Multigrid algorithms for hp-Discontinuous Galerkin discretizations of elliptic problems. SIAM J. Numer. Anal. 53 (2015) 598–618. [CrossRef] [MathSciNet] [Google Scholar]
  8. D.N. Arnold, F. Brezzi, B. Cockburn and L.D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 (2001) 1749–1779. [Google Scholar]
  9. B. Ayuso and L.D. Marini, Discontinuous Galerkin methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal. 47 (2009) 1391–1420. [CrossRef] [MathSciNet] [Google Scholar]
  10. I. Babuška, The finite element method with penalty. Math. Comput. 27 (1973) 221–228. [CrossRef] [MathSciNet] [Google Scholar]
  11. I. Babuška and M. Suri, The h-p version of the finite element method with quasi-uniform meshes. RAIRO Modél. Math. Anal. Numér. 21 (1987) 199–238. [MathSciNet] [Google Scholar]
  12. I. Babuška and M. Suri, The optimal convergence rate of the p-version of the finite element method. SIAM J. Numer. Anal. 24 (1987) 750–776. [CrossRef] [MathSciNet] [Google Scholar]
  13. G.A. Baker, Finite element methods for elliptic equations using nonconforming elements. Math. Comput. 31 (1977) 45–59. [CrossRef] [MathSciNet] [Google Scholar]
  14. F. Bassi, L. Botti and A. Colombo, Agglomeration-based physical frame dG discretizations: An attempt to be mesh free. Math. Models Methods Appl. Sci. 24 (2014) 1495–1539. [Google Scholar]
  15. F. Bassi, L. Botti, A. Colombo, D.A. Di Pietro and P. Tesini, On the flexibility of agglomeration based physical space discontinuous Galerkin discretizations. J. Comput. Phys. 231 (2012) 45–65. [CrossRef] [MathSciNet] [Google Scholar]
  16. F. Bassi, L. Botti, A. Colombo and S. Rebay, Agglomeration based discontinuous Galerkin discretization of the Euler and Navier-Stokes equations. Comput. Fluids 61 (2012) 77–85. [CrossRef] [MathSciNet] [Google Scholar]
  17. S.C. Brenner and J. Zhao, Convergence of multigrid algorithms for interior penalty methods. Appl. Numer. Anal. Comput. Math. 2 (2005) 3–18. [CrossRef] [MathSciNet] [Google Scholar]
  18. S.C. Brenner, J. Cui and L.-Y. Sung, Multigrid methods for the symmetric interior penalty method on graded meshes. Numer. Linear Algebra Appl. 16 (2009) 481–501. [CrossRef] [MathSciNet] [Google Scholar]
  19. A Buffa, T.J.R. Hughes and G Sangalli, Analysis of a multiscale discontinuous Galerkin method for convection-diffusion problems. SIAM J. Numer. Anal. 44 (2006) 1420–1440. [CrossRef] [MathSciNet] [Google Scholar]
  20. A. Cangiani, J. Chapman, E.H. Georgoulis and M. Jensen, On the stability of continuous-discontinuous Galerkin methods for advection-diffusion-reaction problems. J. Sci. Comput. 57 (2013) 313–330. [CrossRef] [MathSciNet] [Google Scholar]
  21. A. Cangiani, E.H. Georgoulis and P. Houston, hp–version discontinuous Galerkin methods on polygonal and polyhedral meshes. Math. Models Methods Appl. Sci. 24 (2014) 2009–2041. [CrossRef] [Google Scholar]
  22. A. Chernov, Optimal convergence estimates for the trace of the polynomial L2-projection operator on a simplex. Math. Comput. 81 (2012) 765–787. [CrossRef] [Google Scholar]
  23. P.G. Ciarlet, The Finite Element Method for Elliptic Problems. Vol. 4 of Stud. Math. Appl. North-Holland Publishing Co., Amsterdam (1978). [Google Scholar]
  24. B. Cockburn, An Introduction to the Discontinuous Galerkin Method for Convection-Dominated Problems. In Advanced numerical approximation of nonlinear hyperbolic equations (Cetraro, 1997). Springer, Berlin (1998) 151–268. [Google Scholar]
  25. B. Cockburn, G.E. Karniadakis and C.-W. Shu., Eds., Discontinuous Galerkin Methods. Theory, Computation and Applications. Papers from the 1st International Symposium held in Newport, RI, May 24–26 1999. In Lect. Notes Comput. Sci. Eng. Springer-Verlag, Berlin (2000). [Google Scholar]
  26. B. Cockburn, B. Dong and J. Guzmán, Optimal convergence of the original DG method for the transport-reaction equation on special meshes. SIAM J. Numer. Anal. 46 (2008) 1250–1265. [CrossRef] [MathSciNet] [Google Scholar]
  27. B. Cockburn, B. Dong, J. Guzmán and J. Qian, Optimal convergence of the original DG method on special meshes for variable transport velocity. SIAM J. Numer. Anal. 48 (2010) 133–146. [CrossRef] [MathSciNet] [Google Scholar]
  28. D.A. Di Pietro and A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods. Vol. 69 of Math. Appl. Springer, Heidelberg (2012). [Google Scholar]
  29. X. Feng and O.A. Karakashian, Two-level additive Schwarz methods for a discontinuous Galerkin approximation of second order elliptic problems. SIAM J. Numer. Anal., 39 (2001) 1343–1365. [CrossRef] [MathSciNet] [Google Scholar]
  30. E.H. Georgoulis, Discontinuous Galerkin methods on shape-regular and anisotropic meshes. D. Phil. thesis, University of Oxford (2003). [Google Scholar]
  31. E.H. Georgoulis, Inverse-type estimates on hp-finite element spaces and applications. Math. Comput. 77 (2008) 201–219. [CrossRef] [Google Scholar]
  32. E.H. Georgoulis and A. Lasis, A note on the design of hp-version interior penalty discontinuous Galerkin finite element methods for degenerate problems. IMA J. Numer. Anal. 26 (2006) 381–390. [CrossRef] [MathSciNet] [Google Scholar]
  33. S. Giani and P. Houston, hp-Adaptive composite discontinuous Galerkin methods for elliptic problems on complicated domains. Num. Meth. Partial Differ. Eqs. 30 (2014) 1342–1367. [CrossRef] [Google Scholar]
  34. P. Houston, C. Schwab and E. Süli, Stabilized hp-finite element methods for first-order hyperbolic problems. SIAM J. Numer. Anal. 37 (2000) 1618–1643. [CrossRef] [MathSciNet] [Google Scholar]
  35. P. Houston and E. Süli, Stabilised hp-finite element approximation of partial differential equations with nonnegative characteristic form. Computing 66 (2001) 99–119. [CrossRef] [MathSciNet] [Google Scholar]
  36. P. Houston, C. Schwab and E. Süli, Discontinuous hp-finite element methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal. 39 (2002) 2133–2163. [CrossRef] [MathSciNet] [Google Scholar]
  37. C. Johnson and J. Pitkäranta, An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. Math. Comput. 46 (1986) 1–26. [CrossRef] [MathSciNet] [Google Scholar]
  38. G. Karypis and V. Kumar, A fast and highly quality multilevel scheme for partitioning irregular graphs. SIAM J. Sci. Comput. 20 (1999) 359–392. [CrossRef] [MathSciNet] [Google Scholar]
  39. C. Lasser and A. Toselli, An overlapping domain decomposition preconditioner for a class of discontinuous Galerkin approximations of advection-diffusion problems. Math. Comput. 72 (2003) 1215–1238. [CrossRef] [Google Scholar]
  40. K. Lipnikov, D. Vassilev and I. Yotov, Discontinuous Galerkin and mimetic finite difference methods for coupled Stokes-Darcy flows on polygonal and polyhedral grids. Numer. Math. (2013) 1–40. [Google Scholar]
  41. R. Muñoz-Sola, Polynomial liftings on a tetrahedron and applications to the hp-version of the finite element method in three dimensions. SIAM J. Numer. Anal. 34 (1997) 282–314. [CrossRef] [MathSciNet] [Google Scholar]
  42. J. Nitsche, Über ein Variationsprinzip zur Lösung von Dirichlet Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Sem. Uni. Hamburg 36 (1971) 9–15. [CrossRef] [Google Scholar]
  43. I. Perugia and D. Schötzau, An hp-analysis of the local discontinuous Galerkin method for diffusion problems. J. Sci. Comput. 17 (2002) 561–571. [CrossRef] [MathSciNet] [Google Scholar]
  44. T.E. Peterson, A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation. SIAM J. Numer. Anal. 28 (1991) 133–140. [CrossRef] [MathSciNet] [Google Scholar]
  45. W.H. Reed and T.R. Hill, Triangular mesh methods for the neutron transport equation. Technical Report LA-UR-73-479, Los Alamos Scientific Laboratory (1973). [Google Scholar]
  46. C. Schwab, p– and hp–Finite Element Methods: Theory and Applications in Solid and Fluid Mechanics. Numerical Mathematics and Scientific Computation. Oxford University Press (1998). [Google Scholar]
  47. E.M. Stein, Singular Integrals and Differentiability Properties of Functions. Princeton, University Press, Princeton, N.J. (1970). [Google Scholar]
  48. C. Talischi, G.H. Paulino, A. Pereira and I.F.M. Menezes, Polymesher: A general-purpose mesh generator for polygonal elements written in Matlab. Struct. Multidisc. Optim. 45 (2012) 309–328,. [CrossRef] [Google Scholar]
  49. R. Verfürth, On the constants in some inverse inequalities for finite element functions. Technical Report 257, University of Bochum (1999). [Google Scholar]
  50. D. Wirasaet, E.J. Kubatko, C.E. Michoski, S. Tanaka, J.J. Westerink and C. Dawson, Discontinuous Galerkin methods with nodal and hybrid modal/nodal triangular, quadrilateral, and polygonal elements for nonlinear shallow water flow. Comput. Methods Appl. Mech. Engrg. 270 (2014) 113–149. [CrossRef] [MathSciNet] [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.

Recommended for you