Hostname: page-component-7bb8b95d7b-2h6rp Total loading time: 0 Render date: 2024-10-05T10:51:58.909Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  01 April 2014

Thirupathi Gudi  and
Johnny Guzmán
Thirupathi Gudi
Affiliation:
Department of Mathematics, Indian Institute of Science, 56002 Bangalore, India.. gudi@math.iisc.ernet.in
Johnny Guzmán
Affiliation:
Division of Applied Mathematics, Brown University, Providence, 02912 RI, USA.; JohnnyGuzman@brown.edu
Get access

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.

Keywords

Contractionadaptive finite elementdiscontinuous Galerkin
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 3 , May 2014 , pp. 753 - 764
Copyright
© EDP Sciences, SMAI 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ainsworth, M., A posteriori error estimation for discontinuous Galerkin finite element approximation. SIAM J. Numer. Anal. 39 (2007) 1777–1798. Google Scholar
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).
Arnold, D.N., An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19 (1982) 742–760. Google Scholar
Arnold, D.N., Brezzi, F., Cockburn, B. and Marini, L.D.. Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 (2002) 1749–1779. Google Scholar
Ayuso, B. and Zikatanov, L.L., Uniformly convergent iterative methods for discontinuous Galerkin discretizations. J. Sci. Comput. 40 (2009) 4–36. Google Scholar
I. Babuška and I. Strouboulis, The Finite Element Method and its Reliability. The Claredon Press, Oxford University Press (2001)
W. Bangerth and R. Rannacher, Adaptive Finite Element Methods for Differential Equations. Birkhåuser Verlag, Basel (2003).
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.
Becker, R., Mao, S. and Shi, Z.C., A convergent nonconforming adaptive finite element method with quasi-optimal complexity. SIAM J. Numer. Anal. 47 (2010) 4639–4659. Google Scholar
R. Becker and S. Mao, Private Communication (2013).
Binev, P., Dahmen, W., and DeVore, R., Adaptive finite element methods with convergence rates. Numer. Math. 97 (2004) 219–268. Google Scholar
Bonito, A. and Nochetto, R.H., Quasi-optimal convergence rate of an adaptive discontinuous Galerkin method. SIAM J. Numer. Anal. 48 (2010) 734–771. Google Scholar
S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, 3rd edn. Springer-Verlag, New York (2008).
Brenner, S.C. and Owens, L., A weakly over-penalized non-symmetric interior penalty method. J. Numer. Anal. Ind. Appl. Math. 2 (2007) 35–48. Google Scholar
Brenner, S.C., Owens, L. and Sung, L.Y., A weakly over-penalized symmetric interior penalty method. Electron. Trans. Numer. Anal. 30 (2008) 107–127. Google Scholar
Brezzi, F., Manzini, G., Marini, D., Pietra, P. and Russo, A., Discontiuous Galerkin Approximations for Elliptic Problems. Numer. Methods Partial Differ. Equ. 16 (2000) 365–378. Google Scholar
Burman, E. and Stamm, B., Low order discontinuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal. 47 (2008) 508–533. Google Scholar
Carstensen, C. and Hoppe, R.H.W., Convergence analysis of an adaptive nonconforming finite element method. Numer. Math. 103 (2006) 251–266. Google Scholar
Carstensen, C. and Hoppe, R., Error reduction and convergence for an adaptive mixed finite element method. Math. Comput. 75 (2006) 1033–1042. Google Scholar
Cascon, J.M., Kreuzer, C., Nochetto, R.H. and Siebert, K.G., Quasi-optimal convergence rate for an adaptive finite element method. SIAM J. Numer. Anal. 46 (2008) 2524–2550. Google Scholar
Chen, L., Holst, M. and Xu, J., Convergence and optimality of adaptive mixed finite element methods. Math. Comput. 78 (2009) 35–53. Google Scholar
Cockburn, B. and Shu, C.-W., The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35 (1998) 2440–2463. Google Scholar
Crouzeix, M. and Raviart, P.A., Conforming and Nonconforming finite element methods for solving the stationary Stokes equations. RAIRO Anal. Numer. 7 (1973) 33–76. Google Scholar
Dörfler, W., A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33 (1996) 1106–1124. Google Scholar
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).
Hoppe, R.H.W., Kanschat, G. and Warburton, T., Convergence analysis ofan adaptive interior penalty discontinuous Galerkin method. SIAM J. Numer. Anal. 47 (2008/09) 534–550. Google Scholar
Karakashian, O. A. and Pascal, F., Convergence of adaptive discontinuous Galerkin approximations of second-order elliptic problems. SIAM J. Numer. Anal. 45 (2007) 641–665. Google Scholar
Mao, S., Zhao, X. and Shi, Z., Convergence of a standard adaptive nonconforming finite element method with optimal complexity. Appl. Numer. Math. 60 (2010) 673–688. Google Scholar
Morin, P., Nochetto, R.H. and Siebert, K.G., Data oscillation and convergence adaptive FEM. SIAM J. Numer. Anal. 38 (2000) 466–488. Google Scholar
Morin, P., Nochetto, R.H. and Siebert, K.G., Convergence of adaptive finite element methods. SIAM Review 44 (2002) 631–658. Google Scholar
Stevenson, R., Optimality of a standard adaptive finite element method. Found. Comput. Math. 7 (2007) 245–269. Google Scholar
R. Verfürth, A Review of A Posteriori Error Estmation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner, Chichester (1995).
Wheeler, M.F., An elliptic collocation-finite-element method with interior penalties. SIAM J. Numer. Anal. 15 (1978) 152–161. Google Scholar