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

HTML articles powered by AMS MathViewer

- by Thirupathi Gudi and Kamana Porwal PDF
- Math. Comp.
**83**(2014), 579-602 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**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 - 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, 1976, pp. 207–216. MR**0440955** - 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**0391502**, 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

## Additional Information

**Thirupathi Gudi**- Affiliation: Department of Mathematics, Indian Institute of Science, Bangalore - 560012, India
- Email: gudi@math.iisc.ernet.in
**Kamana Porwal**- Affiliation: Department of Mathematics, Indian Institute of Science, Bangalore - 560012, India
- Email: kamana@math.iisc.ernet.in
- Received by editor(s): November 1, 2011
- Received by editor(s) in revised form: April 3, 2012, and June 18, 2012
- Published electronically: June 19, 2013
- Additional Notes: The first author’s work is supported by the UGC Center for Advanced Study

The second author’s work is supported in part by the UGC center for Advanced Study and in part by the Council of Scientific and Industrial Research (CSIR) - © Copyright 2013
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp.
**83**(2014), 579-602 - MSC (2010): Primary 65N30, 65N15
- DOI: https://doi.org/10.1090/S0025-5718-2013-02728-7
- MathSciNet review: 3143685