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Adaptive discontinuous Galerkin methods for eigenvalue problems arising in incompressible fluid flows

Cliffe, Andrew; Hall, Edward; Houston, Paul

Authors

Andrew Cliffe Andrew.Cliffe@nottingham.ac.uk

Edward Hall Edward.Hall@nottingham.ac.uk

Paul Houston Paul.Houston@nottingham.ac.uk



Abstract

In this article we consider the a posteriori error estimation and adaptive mesh refinement of discontinuous Galerkin finite element approximations of the hydrodynamic stability problem associated with the incompressible Navier-Stokes equations. Particular attention is given to the reliable error estimation of the eigenvalue problem in channel and pipe geometries. Here, computable a posteriori error bounds are derived based on employing the generalization of the standard Dual-Weighted-Residual approach, originally developed for the estimation of target functionals of the solution, to eigenvalue/stability problems. The underlying analysis consists of constructing both a dual eigenvalue problem and a dual problem for the original base solution. In this way, errors stemming from both the numerical approximation of the original nonlinear flow problem, as well as the underlying linear eigenvalue problem are correctly controlled. Numerical experiments highlighting the practical performance of the proposed a posteriori error indicator on adaptively refined computational meshes are presented.

Journal Article Type Article
Journal SIAM Journal on Scientific Computing
Print ISSN 1064-8275
Electronic ISSN 1064-8275
Publisher Society for Industrial and Applied Mathematics
Peer Reviewed Not Peer Reviewed
APA6 Citation Cliffe, A., Hall, E., & Houston, P. Adaptive discontinuous Galerkin methods for eigenvalue problems arising in incompressible fluid flows. Manuscript submitted for publication
Keywords Incompressible flows, hydrodynamic stability, a posteriori error estimation, adaptivity, discontinuous Galerkin methods
Copyright Statement Copyright information regarding this work can be found at the following address: http://eprints.nottingh.../end_user_agreement.pdf

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Copyright Statement
Copyright information regarding this work can be found at the following address: http://eprints.nottingham.ac.uk/end_user_agreement.pdf





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