Liao, Qifeng
(2010)
*Error estimation and stabilization for low order finite elements.*
Doctoral thesis, Manchester Institute for Mathematical Sciences, The University of Manchester.

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## Abstract

This thesis covers three topicsâ��a posteriori error estimation, mixed finite element ap- proximations for anisotropic meshes and the solution of the time-dependent Navier-Stokes equations using a stabilized Q1 â�� P0 approximation. First, we find effective error estimators for (bi-)quadratic approximations for the dif- fusion problem, and (bi-)quadratic velocity and (bi-)linear pressure mixed approximations for incompressible flow problems. The efficiency and reliability of the error estimators are established in the case of the Stokes problem. Second, since standard inf-sup stable mixed approximations typically become unstable for anisotropic meshes, we devote our attention to a stabilized Q1â��P0 approximation, which is introduced by Kechkar and Silvester [Math. Comp., 58, 1â��10, 1992]. We establish a robust a priori error bound for this stabilized Q1 â�� P0 approximation for anisotropic meshes. Finally, the stabilized Q1 â�� P0 approximation is applied to solving time dependent in- compressible flow problems with an adaptive time stepping method introduced by Kay et al. [SIAM J. Sci. Comput., 32, 111â��128, 2010]. The main contribution of this part is to find the optimal stabilization parameter, which is eventually shown to be inversely proportional to the Reynolds number of the flow.

Item Type: | Thesis (Doctoral) |
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Subjects: | MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis MSC 2010, the AMS's Mathematics Subject Classification > 76 Fluid mechanics |

Depositing User: | professor david silvester |

Date Deposited: | 26 Jan 2011 |

Last Modified: | 20 Oct 2017 14:12 |

URI: | http://eprints.maths.manchester.ac.uk/id/eprint/1571 |

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