A posteriori error estimates and adaptive methods for incompressible viscous flow problems

Abstract

This thesis is focused on the development and numerical justification of a modern computational methodology that provides guaranteed upper bounds of the energy norms of an error. The methodology suggested is based on the so-called functional type a posteriori error estimates. Different linearizations of the Navier-Stokes equations are considered. Namely, estimates of the Stokes problem, the evolutionary Stokes problem and the system with rotation are proposed. For the system with rotation and semi-discrete approximations of the evolutionary Stokes problem, such type of estimates are presented for the first time. For the Stokes problem and the system with rotation, different numerical strategies are implemented. Numerical tests are performed in Cartesian and Cylindrical coordinate system. For the Stokes problem, a posteriori error estimates on a certain subdomain of interest are also tested. It is shown that functional type a posteriori error estimation methods give reliable and robust upper bounds of the error and realistic error indication. The approach suggested allows to construct efficient mesh-adaptive algorithms and provide a guaranteed accuracy for the approximate solutions.