A new error analysis for discontinuous finite element methods for linear elliptic problems

Gudi, Thirupathi

doi:10.1090/S0025-5718-10-02360-4

A new error analysis for discontinuous finite element methods for linear elliptic problems
HTML articles powered by AMS MathViewer

by Thirupathi Gudi PDF

Math. Comp. 79 (2010), 2169-2189 Request permission

Abstract:

The standard a priori error analysis of discontinuous Galerkin methods requires additional regularity on the solution of the elliptic boundary value problem in order to justify the Galerkin orthogonality and to handle the normal derivative on element interfaces that appear in the discrete energy norm. In this paper, a new error analysis of discontinuous Galerkin methods is developed using only the $H^k$ weak formulation of a boundary value problem of order $2k$. This is accomplished by replacing the Galerkin orthogonality with estimates borrowed from a posteriori error analysis and by using a discrete energy norm that is well defined for functions in $H^k$.

References

Mark Ainsworth and J. Tinsley Oden, A posteriori error estimation in finite element analysis, Pure and Applied Mathematics (New York), Wiley-Interscience [John Wiley & Sons], New York, 2000. MR 1885308, DOI 10.1002/9781118032824
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
Garth A. Baker, Finite element methods for elliptic equations using nonconforming elements, Math. Comp. 31 (1977), no. 137, 45–59. MR 431742, DOI 10.1090/S0025-5718-1977-0431742-5
Alan Berger, Ridgway Scott, and Gilbert Strang, Approximate boundary conditions in the finite element method, Symposia Mathematica, Vol. X (Convegno di Analisi Numerica, INDAM, Rome, 1972) Academic Press, London, 1972, pp. 295–313. MR 0403258
Susanne C. Brenner, Two-level additive Schwarz preconditioners for nonconforming finite element methods, Math. Comp. 65 (1996), no. 215, 897–921. MR 1348039, DOI 10.1090/S0025-5718-96-00746-6
Susanne C. Brenner, Convergence of nonconforming multigrid methods without full elliptic regularity, Math. Comp. 68 (1999), no. 225, 25–53. MR 1620215, DOI 10.1090/S0025-5718-99-01035-2
Susanne C. Brenner, Poincaré-Friedrichs inequalities for piecewise $H^1$ functions, SIAM J. Numer. Anal. 41 (2003), no. 1, 306–324. MR 1974504, DOI 10.1137/S0036142902401311
Susanne C. Brenner, Kening Wang, and Jie Zhao, Poincaré-Friedrichs inequalities for piecewise $H^2$ functions, Numer. Funct. Anal. Optim. 25 (2004), no. 5-6, 463–478. MR 2106270, DOI 10.1081/NFA-200042165
Susanne C. Brenner, Discrete Sobolev and Poincaré inequalities for piecewise polynomial functions, Electron. Trans. Numer. Anal. 18 (2004), 42–48. MR 2083293
Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, 3rd ed., Texts in Applied Mathematics, vol. 15, Springer, New York, 2008. MR 2373954, DOI 10.1007/978-0-387-75934-0
Susanne C. Brenner and Li-Yeng Sung, $C^0$ interior penalty methods for fourth order elliptic boundary value problems on polygonal domains, J. Sci. Comput. 22/23 (2005), 83–118. MR 2142191, DOI 10.1007/s10915-004-4135-7
Susanne C. Brenner and Luke Owens, A weakly over-penalized non-symmetric interior penalty method, JNAIAM J. Numer. Anal. Ind. Appl. Math. 2 (2007), no. 1-2, 35–48. MR 2332345
Susanne C. Brenner and Luke Owens, A $W$-cycle algorithm for a weakly over-penalized interior penalty method, Comput. Methods Appl. Mech. Engrg. 196 (2007), no. 37-40, 3823–3832. MR 2340007, DOI 10.1016/j.cma.2007.02.011
Susanne C. Brenner, Luke Owens, and Li-Yeng Sung, A weakly over-penalized symmetric interior penalty method, Electron. Trans. Numer. Anal. 30 (2008), 107–127. MR 2480072
S.C. Brenner, T. Gudi and L.-Y. Sung. An a posteriori error estimator for a quadratic $C^0$ interior penalty method for the biharmonic problem. to appear in IMA J. Numer. Anal., doi:10.1093/imanum/drn057.
Paul Castillo, Bernardo Cockburn, Ilaria Perugia, and Dominik Schötzau, An a priori error analysis of the local discontinuous Galerkin method for elliptic problems, SIAM J. Numer. Anal. 38 (2000), no. 5, 1676–1706. MR 1813251, DOI 10.1137/S0036142900371003
Carsten Carstensen, Thirupathi Gudi, and Max Jensen, A unifying theory of a posteriori error control for discontinuous Galerkin FEM, Numer. Math. 112 (2009), no. 3, 363–379. MR 2501309, DOI 10.1007/s00211-009-0223-9
Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 0520174
Bernardo Cockburn and Chi-Wang Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal. 35 (1998), no. 6, 2440–2463. MR 1655854, DOI 10.1137/S0036142997316712
M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 7 (1973), no. R-3, 33–75. MR 343661
Jim Douglas Jr. and Todd Dupont, Interior penalty procedures for elliptic and parabolic Galerkin methods, Computing methods in applied sciences (Second Internat. Sympos., Versailles, 1975) Lecture Notes in Phys., Vol. 58, Springer, Berlin, 1976, pp. 207–216. MR 0440955
G. Engel, K. Garikipati, T. J. R. Hughes, M. G. Larson, L. Mazzei, and R. L. Taylor, Continuous/discontinuous finite element approximations of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity, Comput. Methods Appl. Mech. Engrg. 191 (2002), no. 34, 3669–3750. MR 1915664, DOI 10.1016/S0045-7825(02)00286-4
Xiaobing Feng and Ohannes A. Karakashian, Fully discrete dynamic mesh discontinuous Galerkin methods for the Cahn-Hilliard equation of phase transition, Math. Comp. 76 (2007), no. 259, 1093–1117. MR 2299767, DOI 10.1090/S0025-5718-07-01985-0
Xiaobing Feng and Ohannes A. Karakashian, Two-level non-overlapping Schwarz preconditioners for a discontinuous Galerkin approximation of the biharmonic equation, J. Sci. Comput. 22/23 (2005), 289–314. MR 2142199, DOI 10.1007/s10915-004-4141-9
E. H. Georgoulis P. Houston, and J. Virtanen. An a posteriori error indicator for discontinuous Galerkin approximations of fourth order elliptic problems. Nottingham eprint.
P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 775683
Thirupathi Gudi, Neela Nataraj, and Amiya K. Pani, Mixed discontinuous Galerkin finite element method for the biharmonic equation, J. Sci. Comput. 37 (2008), no. 2, 139–161. MR 2453216, DOI 10.1007/s10915-008-9200-1
Paul Houston, Dominik Schötzau, and Thomas P. Wihler, Energy norm a posteriori error estimation of $hp$-adaptive discontinuous Galerkin methods for elliptic problems, Math. Models Methods Appl. Sci. 17 (2007), no. 1, 33–62. MR 2290408, DOI 10.1142/S0218202507001826
Ohannes A. Karakashian and Frederic Pascal, A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems, SIAM J. Numer. Anal. 41 (2003), no. 6, 2374–2399. MR 2034620, DOI 10.1137/S0036142902405217
R. B. Kellogg, Singularities in interface problems, Numerical Solution of Partial Differential Equations, II (SYNSPADE 1970) (Proc. Sympos., Univ. of Maryland, College Park, Md., 1970) Academic Press, New York, 1971, pp. 351–400. MR 0289923
L.S.D. Morley. The triangular equilibrium problem in the solution of plate bending problems. Aero. Quart., 19:149–169, 1968.
Igor Mozolevski, Endre Süli, and Paulo R. Bösing, $hp$-version a priori error analysis of interior penalty discontinuous Galerkin finite element approximations to the biharmonic equation, J. Sci. Comput. 30 (2007), no. 3, 465–491. MR 2295480, DOI 10.1007/s10915-006-9100-1
L. Owens. Multigrid Methods for Weakly Over-Penalized Interior penalty Methods. Ph.D. thesis, Department of Mathematics, University of South Corolina, 2007.
S. Prudhomme, F. Pascal and J.T. Oden. Review of error estimation for discontinuous Galerkin methods. TICAM Report 00-27, October 17, 2000.
Béatrice Rivière, Mary F. Wheeler, and Vivette Girault, A priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems, SIAM J. Numer. Anal. 39 (2001), no. 3, 902–931. MR 1860450, DOI 10.1137/S003614290037174X
R. Verfürth, A posteriori error estimation and adaptive mesh-refinement techniques, Proceedings of the Fifth International Congress on Computational and Applied Mathematics (Leuven, 1992), 1994, pp. 67–83. MR 1284252, DOI 10.1016/0377-0427(94)90290-9
R. Verfürth. A Review of A Posteriori Error Estmation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner, Chichester, 1995.
Mary Fanett Wheeler, An elliptic collocation-finite element method with interior penalties, SIAM J. Numer. Anal. 15 (1978), no. 1, 152–161. MR 471383, DOI 10.1137/0715010