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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)
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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