A posteriori error estimates for variational problems in the theory of viscous fluids

Abstract

The papers included in the thesis are focused on functional type a posteriori error estimates for the Stokes problem, the Stokes problem with friction type boundary conditions, the Oseen problem, and the anti-plane Bingham problem. In the summary of the thesis we consider only the Oseen problem. The papers present and justify special forms of these estimates which are suitable for the approximations generated by the Uzawa algorithm. The estimates are of two main types. Estimates of the first type use exact solutions obtained on the steps of the Uzawa algorithm. They show how errors encompassed in Uzawa approximations behave and have mainly theoretical meaning. Estimates of the second type operate only with approximations (e.g. finite element solutions). Therefore, they are fully computable. In the thesis it is shown that estimates of this type indeed provide realistic evaluation of errors for finite element approximations of problems associated with viscous incompressible fluids.