Commun. Comput. Phys., 8 (2010), pp. 845-865.


Adaptivity and A Posteriori Error Control for Bifurcation Problems I: The Bratu Problem

K. Andrew Cliffe 1, Edward J. C. Hall 1, Paul Houston 1*, Eric T. Phipps 2, Andrew G. Salinger 2

1 School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK.
2 Computer Science Research Institute, Sandia National Laboratories, Albuquerque, New Mexico, USA.

Received 29 July 2009; Accepted (in revised version) 12 February 2010
Available online 17 May 2010
doi:10.4208/cicp.290709.120210a

Abstract

This article is concerned with the numerical detection of bifurcation points of nonlinear partial differential equations as some parameter of interest is varied. In particular, we study in detail the numerical approximation of the Bratu problem, based on exploiting the symmetric version of the interior penalty discontinuous Galerkin finite element method. A framework for a posteriori control of the discretization error in the computed critical parameter value is developed based upon the application of the dual weighted residual (DWR) approach. Numerical experiments are presented to highlight the practical performance of the proposed a posteriori error estimator.

AMS subject classifications: 37G10, 37M20, 65N12, 65N15, 65N30

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Key words: Bifurcation theory, Bratu problem, a posteriori error estimation, adaptivity, discontinuous Galerkin methods.

*Corresponding author.
Email: Andrew.Cliffe@nottingham.ac.uk (K. A. Cliffe), Edward.Hall@nottingham.ac.uk (E. J. C. Hall), Paul.Houston@nottingham.ac.uk (P. Houston), etphipp@sandia.gov (E. T. Phipps), agsalin@sandia.gov (A. G. Salinger)
 

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