Error estimation and stabilization for low order finite elements

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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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 (Doctoral)
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

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