Adaptivity and a Posteriori Error Control for Bifurcation Problems I: the Bratu Problem

Abstract

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.

An $hp$-adaptive Newton-discontinuous-Galerkin finite element approach for semilinear elliptic boundary value problems

“…In the one-dimensional setting, an analytical expression for λ c is available, cf. [7,13,17]; for the two-dimensional case, calculations have revealed that λ c = 6.808124423 ( c = 0.146883332) to 9 decimal places, see [17,44,45], and the references cited therein.…”

Adaptive Discontinuous Galerkin Methods for Eigenvalue Problems Arising in Incompressible Fluid Flows

“…It is worth pointing out that numerical experiments indicate that the resulting error indicators using either of these definitions for J(·) lead to almost identical results. However, the latter definition is more pertinent in the context of parameter estimation in bifurcation problems; see, for example, [9].…”

“…Thereby, upon linearization of the unsteady Navier-Stokes equations (2.1)-(2.2), we obtain the following eigenvalue problem for the pair {λ m , (u m , p m )}: 9) subject to homogeneous Dirichlet and Neumann conditions As is customary, we also enforce the (scaling) condition u m 0 = 1, where · 0 denotes the L 2 (Ω)-norm. We shall now refer to (2.8)-(2.12) as the primal eigenvalue problem.…”

Adaptivity and a Posteriori Error Control for Bifurcation Problems III: Incompressible Fluid Flow in Open Systems with O(2) Symmetry

“…To this end, we are interested in numerically estimating the critical Reynolds number Re, at which a (pitchfork) bifurcation point first occurs; a review of techniques for bifurcation detection can be found in Cliffe et al [13], for example. The work in this article expands upon our recent work in [12] and [10] to include problems whose geometries exhibit both rotational and reflectional symmetry. The detection of bifurcation points in this setting is now well understood, for example, see Golubitsky and Schaeffer [19].…”