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

Guzmán, Johnny

doi:10.1090/S0025-5718-06-01823-0

Pointwise error estimates for discontinuous Galerkin methods with lifting operators for elliptic problems
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by Johnny Guzmán PDF

Math. Comp. 75 (2006), 1067-1085 Request permission

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.

References

Douglas N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal. 19 (1982), no. 4, 742–760. MR 664882, DOI 10.1137/0719052
Douglas N. Arnold, Franco Brezzi, Bernardo Cockburn, and L. Donatella Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal. 39 (2001/02), no. 5, 1749–1779. MR 1885715, DOI 10.1137/S0036142901384162
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.
F. Brezzi, G. Manzini, D. Marini, P. Pietra, and A. Russo, Discontinuous Galerkin approximations for elliptic problems, Numer. Methods Partial Differential Equations 16 (2000), no. 4, 365–378. MR 1765651, DOI 10.1002/1098-2426(200007)16:4<365::AID-NUM2>3.0.CO;2-Y
H. Chen, Local error estimates of mixed discontinuous Galerkin methods for elliptic problems, J. Numer. Math. 12 (2004), no. 1, 1–21. MR 2039367, DOI 10.1163/1569395041172926
Zhangxin Chen and Hongsen Chen, Pointwise error estimates of discontinuous Galerkin methods with penalty for second-order elliptic problems, SIAM J. Numer. Anal. 42 (2004), no. 3, 1146–1166. MR 2113680, DOI 10.1137/S0036142903421527
Jim Douglas Jr., Todd Dupont, and Lars Wahlbin, The stability in $L^{q}$ of the $L^{2}$-projection into finite element function spaces, Numer. Math. 23 (1974/75), 193–197. MR 383789, DOI 10.1007/BF01400302
G. Kanschat and R. Rannacher, Local error analysis of the interior penalty discontinuous Galerkin method for second order elliptic problems, J. Numer. Math. 10 (2002), no. 4, 249–274. MR 1954085, DOI 10.1515/JNMA.2002.249
Joachim A. Nitsche and Alfred H. Schatz, Interior estimates for Ritz-Galerkin methods, Math. Comp. 28 (1974), 937–958. MR 373325, DOI 10.1090/S0025-5718-1974-0373325-9
Alfred H. Schatz, An observation concerning Ritz-Galerkin methods with indefinite bilinear forms, Math. Comp. 28 (1974), 959–962. MR 373326, DOI 10.1090/S0025-5718-1974-0373326-0
A. H. Schatz and L. B. Wahlbin, Interior maximum norm estimates for finite element methods, Math. Comp. 31 (1977), no. 138, 414–442. MR 431753, DOI 10.1090/S0025-5718-1977-0431753-X
Alfred H. Schatz, Pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids. I. Global estimates, Math. Comp. 67 (1998), no. 223, 877–899. MR 1464148, DOI 10.1090/S0025-5718-98-00959-4
Lars B. Wahlbin, Local behavior in finite element methods, Handbook of numerical analysis, Vol. II, Handb. Numer. Anal., II, North-Holland, Amsterdam, 1991, pp. 353–522. MR 1115238