A posteriori error control of discontinuous Galerkin methods for elliptic obstacle problems

Gudi, Thirupathi; Porwal, Kamana

doi:10.1090/S0025-5718-2013-02728-7

A posteriori error control of discontinuous Galerkin methods for elliptic obstacle problems
HTML articles powered by AMS MathViewer

by Thirupathi Gudi and Kamana Porwal;

Math. Comp. 83 (2014), 579-602

DOI: https://doi.org/10.1090/S0025-5718-2013-02728-7

Published electronically: June 19, 2013

PDF | Request permission

Abstract:

In this article, we derive an a posteriori error estimator for various discontinuous Galerkin (DG) methods that are proposed in (Wang, Han and Cheng, SIAM J. Numer. Anal., 48:708–733, 2010) for an elliptic obstacle problem. Using a key property of DG methods, we perform the analysis in a general framework. The error estimator we have obtained for DG methods is comparable with the estimator for the conforming Galerkin (CG) finite element method. In the analysis, we construct a non-linear smoothing function mapping DG finite element space to CG finite element space and use it as a key tool. The error estimator consists of a discrete Lagrange multiplier associated with the obstacle constraint. It is shown for non-over-penalized DG methods that the discrete Lagrange multiplier is uniformly stable on non-uniform meshes. Finally, numerical results demonstrating the performance of the error estimator are presented.

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
Kendall Atkinson and Weimin Han, Theoretical numerical analysis, 3rd ed., Texts in Applied Mathematics, vol. 39, Springer, Dordrecht, 2009. A functional analysis framework. MR 2511061, DOI 10.1007/978-1-4419-0458-4
Ivo Babuška and Miloš Zlámal, Nonconforming elements in the finite element method with penalty, SIAM J. Numer. Anal. 10 (1973), 863–875. MR 345432, DOI 10.1137/0710071
S. Bartels and C. Carstensen, Averaging techniques yield reliable a posteriori finite element error control for obstacle problems, Numer. Math. 99 (2004), no. 2, 225–249. MR 2107431, DOI 10.1007/s00211-004-0553-6
F. Bassi, S. Rebay, G. Mariotti, S. Pedinotti, and M. Savini. A high-order accurate discontinuous finite element method for inviscid and viscous 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, pp. 99-108.
Dietrich Braess, A posteriori error estimators for obstacle problems—another look, Numer. Math. 101 (2005), no. 3, 415–421. MR 2194822, DOI 10.1007/s00211-005-0634-1
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 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, Luke Owens, and Li-Yeng Sung, A weakly over-penalized symmetric interior penalty method, Electron. Trans. Numer. Anal. 30 (2008), 107–127. MR 2480072
Franco Brezzi, William W. Hager, and P.-A. Raviart, Error estimates for the finite element solution of variational inequalities, Numer. Math. 28 (1977), no. 4, 431–443. MR 448949, DOI 10.1007/BF01404345
F. Brezzi, G. Manzini, D. Marini, P. Pietra and A. Russo. Discontinuous finite elements for diffusion problems, in Atti Convegno in onore di F. Brioschi (Milan, 1997), Istituto Lombardo, Accademia di Scienze e Lettere, Milan, Italy, 1999, pp. 197-217.
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
Dietrich Braess, Carsten Carstensen, and Ronald H. W. Hoppe, Convergence analysis of a conforming adaptive finite element method for an obstacle problem, Numer. Math. 107 (2007), no. 3, 455–471. MR 2336115, DOI 10.1007/s00211-007-0098-6
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
Zhiming Chen and Ricardo H. Nochetto, Residual type a posteriori error estimates for elliptic obstacle problems, Numer. Math. 84 (2000), no. 4, 527–548. MR 1742264, DOI 10.1007/s002110050009
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 520174
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
Willy Dörfler, A convergent adaptive algorithm for Poisson’s equation, SIAM J. Numer. Anal. 33 (1996), no. 3, 1106–1124. MR 1393904, DOI 10.1137/0733054
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-New York, 1976, pp. 207–216. MR 440955
Roland Glowinski, Numerical methods for nonlinear variational problems, Scientific Computation, Springer-Verlag, Berlin, 2008. Reprint of the 1984 original. MR 2423313
Thirupathi Gudi, A new error analysis for discontinuous finite element methods for linear elliptic problems, Math. Comp. 79 (2010), no. 272, 2169–2189. MR 2684360, DOI 10.1090/S0025-5718-10-02360-4
Thirupathi Gudi, Neela Nataraj, and Amiya K. Pani, $hp$-discontinuous Galerkin methods for strongly nonlinear elliptic boundary value problems, Numer. Math. 109 (2008), no. 2, 233–268. MR 2385653, DOI 10.1007/s00211-008-0137-y
Richard S. Falk, Error estimates for the approximation of a class of variational inequalities, Math. Comput. 28 (1974), 963–971. MR 391502, DOI 10.1090/S0025-5718-1974-0391502-8
M. Hintermüller, K. Ito, and K. Kunisch, The primal-dual active set strategy as a semismooth Newton method, SIAM J. Optim. 13 (2002), no. 3, 865–888 (2003). MR 1972219, DOI 10.1137/S1052623401383558
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
David Kinderlehrer and Guido Stampacchia, An introduction to variational inequalities and their applications, Classics in Applied Mathematics, vol. 31, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. Reprint of the 1980 original. MR 1786735, DOI 10.1137/1.9780898719451
Ricardo H. Nochetto, Tobias von Petersdorff, and Chen-Song Zhang, A posteriori error analysis for a class of integral equations and variational inequalities, Numer. Math. 116 (2010), no. 3, 519–552. MR 2684296, DOI 10.1007/s00211-010-0310-y
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
Vidar Thomée, Galerkin finite element methods for parabolic problems, 2nd ed., Springer Series in Computational Mathematics, vol. 25, Springer-Verlag, Berlin, 2006. MR 2249024
Andreas Veeser, Efficient and reliable a posteriori error estimators for elliptic obstacle problems, SIAM J. Numer. Anal. 39 (2001), no. 1, 146–167. MR 1860720, DOI 10.1137/S0036142900370812
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 Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner, Chichester, 1995.
Lie-heng Wang, On the quadratic finite element approximation to the obstacle problem, Numer. Math. 92 (2002), no. 4, 771–778. MR 1935809, DOI 10.1007/s002110100368
Fei Wang, Weimin Han, and Xiao-Liang Cheng, Discontinuous Galerkin methods for solving elliptic variational inequalities, SIAM J. Numer. Anal. 48 (2010), no. 2, 708–733. MR 2670002, DOI 10.1137/09075891X
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