Journal of Mathematical Sciences, Vol. 142, No. 1, 2007 A POSTERIORI ERROR ESTIMATES FOR THE GENERALIZED STOKES PROBLEM S. Repin V. A. Steklov Institute of Mathematics in St.-Petersburg, Russia repin@pdmi.ras.ru R. Stenberg Helsinki University of Technology, Institute of Mathematics, Finland rstenber@cc.hut.fi UDC 517.95 The paper concerns a posteriori estimates of functional type for the difference between exact and approximate solutions to a generalized Stokes problem. The estimates are derived by transformations of the basic integral identity defining a generalized solution to the problem using the method suggested by the first author. The estimates obtained can be classified into two types. Estimates of the first type are valid only for solenoidal functions, while estimates of the second type are applicable for any functions that belong to the energy space of the respective problem and satisfy the boundary conditions. In the second case, the estimates include an additional penalty term with a multiplier defined by the constant in the Ladyzhenskaya–Babuˇ ska–Brezzi condition. It is proved that a posteriori estimates for the velocity field yield computable estimates of the difference between exact and approximate pressure functions in the L 2 -norm. It is shown that the estimates provide sharp upper and lower bounds of the error and their practical computation requires to solve only finite–dimensional problems. Bibliography: 34 titles. 1. Generalized Stokes Problem Let Ω ∈ R n (n =2, 3) be a bounded domain. The generalized Stokes problem is to find a vector– valued function u (velocity) and a scalar function p (pressure) that satisfy the system Translated from Problemy Matematicheskogo Analiza, No. 34, 2006, pp. 89–101. c 1828 1072-3374/07/1421-1828  2007 Springer Science+Business Media, Inc.