Estimates of the deviations from the exact solutions for variational inequalities describing the stationary flow of certain viscous incompressible fluids

Abstract

This paper is concerned with computable and guaranteed upper bounds of the difference between exact solutions of variational inequalities arising in the theory of viscous fluids and arbitrary approximations in the corresponding energy space. Such estimates (also called error majorants of functional type) have been derived for the considered class of nonlinear boundary value problems in [11] with the help of variational methods based on duality theory from convex analysis. In the present paper it is shown that error majorants can be derived in a different way by certain transformations of the variational inequalities that define generalized solutions. The error bounds derived by this techniques for the velocity function differ from those obtained by the variational method. These estimates involve only global constants coming from Korn and Friedrichs type inequalities, which are not difficult to evaluate in case of Dirichlet boundary conditions. For the case of mixed boundary conditions, we also derive another form of the estimate which contains only one constant coming from the following assertion: the L^{2} norm of a vector valued function from H^{1}(\Omega) in the factor-space generated by the equivalence with respect to rigid motions is bounded by the L^{2} norm of the symmetric part of the gradient tensor. Since for some ”simple” domains like squares or cubes, the constants in this inequality can be found analytically (or numerically), we obtain a unified form of an error majorant for any domain that admits a decomposition into such subdomains.