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Pointwise error estimates for discontinuous Galerkin methods with lifting operators for elliptic problems

Author: Johnny Guzmán
Journal: Math. Comp. 75 (2006), 1067-1085
MSC (2000): Primary 65N30, 65N15
Published electronically: March 3, 2006
MathSciNet review: 2219019
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Abstract | References | Similar Articles | Additional Information

Abstract: In this article, we prove some weighted pointwise estimates for three discontinuous Galerkin methods with lifting operators appearing in their corresponding bilinear forms. We consider a Dirichlet problem with a general second-order elliptic operator.

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  • 1. D.N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal., 19(1982), 742-760. MR 0664882 (83f:65173)
  • 2. 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(2002), 1749-1779. MR 1885715 (2002k:65183)
  • 3. F. Bassi, S. Rebay, G. Marrioti, S. Pendinotti, and M. Savini, A high-order accurate discontinuous finite element method for inviscid and viscuous turbomachinery flows, In Proceedings of 2nd European Conference on Turbomachinery, Fluid Dynamics and Thermodynamics (R. Decuypere and G. Dibelius, eds.), Technologisch Instituut, Antwerpen, Belgium, 1997, 99-108.
  • 4. F. Brezzi, G. Manzini, D. Marini, P. Pietra, and A. Russo, Discontinuous Galerkin approximations for elliptic problems, Numer. Methods Partial Differential Equations, 16(2000), 365-378. MR 1765651 (2001e:65178)
  • 5. H. Chen, Local error estimates of mixed discontinuous Galerkin methods for elliptic problems, J. Numer. Math., 12(2004), 1-22. MR 2039367 (2005a:65118)
  • 6. H. Chen and Z. Chen, Pointwise estimates of discontinuous Galerkin methods with penalty for second order elliptic problems, SIAM J. Numer. Anal., 42(2004), 1146-1166. MR 2113680
  • 7. J. Douglas, Jr., T. Dupont and L. Wahlbin, The stability in $ L^q$ of the $ L^2$-projection into finite element function spaces, Numer. Math., 23(1975), 193-197. MR 0383789 (52:4669)
  • 8. G. Kanschat and R. Rannacher, Local error analysis of the interior penalty discontinuous Galerkin method for second order elliptic problems, J. Numer. Math., 10(2003), 249-274. MR 1954085 (2004f:65187)
  • 9. J.A. Nitsche and A.H. Schatz, Interior estimates for Ritz-Galerkin methods, Math. Comp., 28(1974), 937-958. MR 0373325 (51:9525)
  • 10. A.H. Schatz, An observation concerning Ritz-Galerkin methods with indefinite bilinear forms, Math. Comp., 28(1974), 959-962. MR 0373326 (51:9526)
  • 11. A.H. Schatz and L.B. Wahlbin, Interior maximum norm estimates for finite element methods, Math. Comp., 31(1977), 414-442. MR 0431753 (55:4748)
  • 12. A.H. Schatz, Pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids: Part I. Global Estimates, Math. Comp., 67 (1998), 877-899. MR 1464148 (98j:65082)
  • 13. L.B. Wahlbin, Local behavior in finite element methods, In Handbook of Numerical Analysis, Volume II, 353-522, North-Holland, Amsterdam, 1991. MR 1115238

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Additional Information

Johnny Guzmán
Affiliation: Center for Applied Mathematics, Cornell University, 657 Rhodes Hall, Ithaca, New York 14853

Keywords: Finite elements, discontinuous Galerkin
Received by editor(s): June 27, 2004
Received by editor(s) in revised form: April 19, 2005
Published electronically: March 3, 2006
Additional Notes: The author was supported by a Ford Foundation Fellowship and a Cornell-Sloan Fellowship
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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