Abstract:We derive computable bounds of deviations from the exact solution of the stationary Oseen problem. They are applied to approximations generated by the Uzawa iteration method. Also, we derive an advanced form of the estimate, which takes into account approximation errors arising due to discretization of the boundary value problem, generated by the main step of the Uzawa method. Numerical tests con rm our theoretical results and show practical applicability of the estimates.

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The paper is concerned with fully guaranteed and computable bounds of errors generated by Uzawa type methods for variational problems in the theory of visco-plastic fluids. The respective estimates have two forms. The first form contains global constants (such as the constant in the Friedrichs inequality for the respective domain), and the second one is based upon decomposition of the domain into a collection of subdomains and uses local constants associated with subdomains.

We deduce a posteriori error estimates of functional type for the stationary Stokes problem with slip and leak boundary conditions. The derived error majorants do not contain mesh dependent constants and are valid for a wide class of energy admissible approximations that satisfy the Dirichlet boundary condition on a part of the boundary. Different forms of error majorants contain global constants associated with Poincaré type inequalities or the stability (LBB) condition for the Stokes problem or constants associated with subdomains (if a domain decomposition is applied). It is proved that the majorants are guaranteed and vanish if and only if the functions entering them coincide with the respective exact solutions. Bibliography: 38 titles. Illustrations: 1 figure.