Convergence analysis of the lowest order weakly penalized adaptive discontinuous Galerkin methods

Thirupathi Gudi; Johnny Guzmán

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2014)

  • Volume: 48, Issue: 3, page 753-764
  • ISSN: 0764-583X

Abstract

top
In this article, we prove convergence of the weakly penalized adaptive discontinuous Galerkin methods. Unlike other works, we derive the contraction property for various discontinuous Galerkin methods only assuming the stabilizing parameters are large enough to stabilize the method. A central idea in the analysis is to construct an auxiliary solution from the discontinuous Galerkin solution by a simple post processing. Based on the auxiliary solution, we define the adaptive algorithm which guides to the convergence of adaptive discontinuous Galerkin methods.

How to cite

top

Gudi, Thirupathi, and Guzmán, Johnny. "Convergence analysis of the lowest order weakly penalized adaptive discontinuous Galerkin methods." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 48.3 (2014): 753-764. <http://eudml.org/doc/273197>.

@article{Gudi2014,
abstract = {In this article, we prove convergence of the weakly penalized adaptive discontinuous Galerkin methods. Unlike other works, we derive the contraction property for various discontinuous Galerkin methods only assuming the stabilizing parameters are large enough to stabilize the method. A central idea in the analysis is to construct an auxiliary solution from the discontinuous Galerkin solution by a simple post processing. Based on the auxiliary solution, we define the adaptive algorithm which guides to the convergence of adaptive discontinuous Galerkin methods.},
author = {Gudi, Thirupathi, Guzmán, Johnny},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {contraction; adaptive finite element; discontinuous Galerkin; discontinuous Galerkin method; adaptive finite elements; marking strategy},
language = {eng},
number = {3},
pages = {753-764},
publisher = {EDP-Sciences},
title = {Convergence analysis of the lowest order weakly penalized adaptive discontinuous Galerkin methods},
url = {http://eudml.org/doc/273197},
volume = {48},
year = {2014},
}

TY - JOUR
AU - Gudi, Thirupathi
AU - Guzmán, Johnny
TI - Convergence analysis of the lowest order weakly penalized adaptive discontinuous Galerkin methods
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2014
PB - EDP-Sciences
VL - 48
IS - 3
SP - 753
EP - 764
AB - In this article, we prove convergence of the weakly penalized adaptive discontinuous Galerkin methods. Unlike other works, we derive the contraction property for various discontinuous Galerkin methods only assuming the stabilizing parameters are large enough to stabilize the method. A central idea in the analysis is to construct an auxiliary solution from the discontinuous Galerkin solution by a simple post processing. Based on the auxiliary solution, we define the adaptive algorithm which guides to the convergence of adaptive discontinuous Galerkin methods.
LA - eng
KW - contraction; adaptive finite element; discontinuous Galerkin; discontinuous Galerkin method; adaptive finite elements; marking strategy
UR - http://eudml.org/doc/273197
ER -

References

top
  1. [1] M. Ainsworth, A posteriori error estimation for discontinuous Galerkin finite element approximation. SIAM J. Numer. Anal.39 (2007) 1777–1798. Zbl1151.65083MR2338409
  2. [2] M. Ainsworth and J.T. Oden, A posteriori error estimation in finite element analysis. Pure and Applied Mathematics. Wiley-Interscience, John Wiley & Sons, New York (2000). Zbl1008.65076MR1885308
  3. [3] D.N. Arnold, An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal.19 (1982) 742–760. Zbl0482.65060MR664882
  4. [4] 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. Zbl1008.65080MR1885715
  5. [5] B. Ayuso and L.L. Zikatanov, Uniformly convergent iterative methods for discontinuous Galerkin discretizations. J. Sci. Comput.40 (2009) 4–36. Zbl1203.65242MR2511726
  6. [6] I. Babuška and I. Strouboulis, The Finite Element Method and its Reliability. The Claredon Press, Oxford University Press (2001) Zbl0995.65501MR1857191
  7. [7] W. Bangerth and R. Rannacher, Adaptive Finite Element Methods for Differential Equations. Birkhåuser Verlag, Basel (2003). Zbl1020.65058MR1960405
  8. [8] F. Bassi, S. Rebay, G. Mariotti, S. Pedinotti, and M. Savini, A higher order accurate discontinuous finite element method for inviscid and viscous turbomachinery flows, in Proc. of 2nd European Conference on Turbomachinery, Fluid Dynamics and Thermodynamics, edited by R. Decuypere and G. Dilbelius, Technologisch Instituut, Antewerpen, Belgium (1997) 99–108. 
  9. [9] R. Becker, S. Mao and Z.C. Shi, A convergent nonconforming adaptive finite element method with quasi-optimal complexity. SIAM J. Numer. Anal.47 (2010) 4639–4659. Zbl1208.65154MR2595052
  10. [10] R. Becker and S. Mao, Private Communication (2013). 
  11. [11] P. Binev, W. Dahmen, and R. DeVore, Adaptive finite element methods with convergence rates. Numer. Math.97 (2004) 219–268. Zbl1063.65120MR2050077
  12. [12] A. Bonito and R.H. Nochetto, Quasi-optimal convergence rate of an adaptive discontinuous Galerkin method. SIAM J. Numer. Anal.48 (2010) 734–771. Zbl1254.65120MR2670003
  13. [13] S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, 3rd edn. Springer-Verlag, New York (2008). Zbl0804.65101MR2373954
  14. [14] S.C. Brenner and L. Owens, A weakly over-penalized non-symmetric interior penalty method. J. Numer. Anal. Ind. Appl. Math.2 (2007) 35–48. Zbl1145.65095MR2332345
  15. [15] S.C. Brenner, L. Owens and L.Y. Sung, A weakly over-penalized symmetric interior penalty method. Electron. Trans. Numer. Anal.30 (2008) 107–127. Zbl1171.65077MR2480072
  16. [16] F. Brezzi, G. Manzini, D. Marini, P. Pietra and A. Russo, Discontiuous Galerkin Approximations for Elliptic Problems. Numer. Methods Partial Differ. Equ.16 (2000) 365–378. Zbl0957.65099MR1765651
  17. [17] E. Burman and B. Stamm, Low order discontinuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal.47 (2008) 508–533. Zbl1190.65170MR2475950
  18. [18] C. Carstensen and R.H.W. Hoppe, Convergence analysis of an adaptive nonconforming finite element method. Numer. Math.103 (2006) 251–266. Zbl1101.65102MR2222810
  19. [19] C. Carstensen and R. Hoppe, Error reduction and convergence for an adaptive mixed finite element method. Math. Comput.75 (2006) 1033–1042. Zbl1094.65112MR2219017
  20. [20] J.M. Cascon, C. Kreuzer, R.H. Nochetto and K.G. Siebert, Quasi-optimal convergence rate for an adaptive finite element method. SIAM J. Numer. Anal.46 (2008) 2524–2550. Zbl1176.65122MR2421046
  21. [21] L. Chen, M. Holst and J. Xu, Convergence and optimality of adaptive mixed finite element methods. Math. Comput.78 (2009) 35–53. Zbl1198.65211MR2448696
  22. [22] B. Cockburn and C.-W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal.35 (1998) 2440–2463. Zbl0927.65118MR1655854
  23. [23] M. Crouzeix and P.A. Raviart, Conforming and Nonconforming finite element methods for solving the stationary Stokes equations. RAIRO Anal. Numer.7 (1973) 33–76. Zbl0302.65087MR343661
  24. [24] W. Dörfler, A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal.33 (1996) 1106–1124. Zbl0854.65090MR1393904
  25. [25] J. Douglas, Jr. and T. Dupont, Interior penalty procedures for elliptic and parabolic Galerkin methods. In vol. 58. Lect. Notes Phys. Springer-Verlag, Berlin (1976). MR440955
  26. [26] R.H.W. Hoppe, G. Kanschat and T. Warburton, Convergence analysis ofan adaptive interior penalty discontinuous Galerkin method. SIAM J. Numer. Anal. 47 (2008/09) 534–550. Zbl1189.65274MR2475951
  27. [27] O. A. Karakashian and F. Pascal, Convergence of adaptive discontinuous Galerkin approximations of second-order elliptic problems. SIAM J. Numer. Anal.45 (2007) 641–665. Zbl1140.65083MR2300291
  28. [28] S. Mao, X. Zhao and Z. Shi, Convergence of a standard adaptive nonconforming finite element method with optimal complexity. Appl. Numer. Math.60 (2010) 673–688. Zbl1202.65147MR2646469
  29. [29] P. Morin, R.H. Nochetto and K.G. Siebert, Data oscillation and convergence adaptive FEM. SIAM J. Numer. Anal.38 (2000) 466–488. Zbl0970.65113MR1770058
  30. [30] P. Morin, R.H. Nochetto and K.G. Siebert, Convergence of adaptive finite element methods. SIAM Review44 (2002) 631–658. Zbl1016.65074MR1980447
  31. [31] R. Stevenson, Optimality of a standard adaptive finite element method. Found. Comput. Math.7 (2007) 245–269. Zbl1136.65109MR2324418
  32. [32] R. Verfürth, A Review of A Posteriori Error Estmation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner, Chichester (1995). Zbl0853.65108
  33. [33] M.F. Wheeler, An elliptic collocation-finite-element method with interior penalties. SIAM J. Numer. Anal.15 (1978) 152–161. Zbl0384.65058MR471383

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.