## 2011.14: Error estimation and stabilization for low order finite elements

2011.14:
Qifeng Liao
(2010)
*Error estimation and stabilization for low order finite elements.*
PhD thesis, Manchester Institute for Mathematical Sciences, The University of Manchester.

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## Abstract

This thesis covers three topics—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 − 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 Q1−P0 approximation, which is introduced by Kechkar and Silvester [Math. Comp., 58, 1–10, 1992]. We establish a robust a priori error bound for this stabilized Q1 − P0 approximation for anisotropic meshes. Finally, the stabilized Q1 − 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, 111–128, 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 (PhD) |
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Subjects: | MSC 2000 > 65 Numerical analysis MSC 2000 > 76 Fluid mechanics |

MIMS number: | 2011.14 |

Deposited By: | professor david silvester |

Deposited On: | 26 January 2011 |

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