A Posteriori Identitiesfor the Generalised Stokes Problem

In the paper, functional identities for the difference between a given function and the exact solution of the generalized Stokes problem are derived. The restrictions on the form of such a function are minimal. Actually, they are reduced to the requirement that it must belong to the same function class as the solution of the problem. The left part of the identity represents a weighted sum of norms and characterizes deviations from the exact velocity and stresses fields. The right part includes a number of terms. Some of them can be directly computed from the problem data and known approximate solutions. Other terms contain unknown functions but can be efficiently estimated. Hence the identity is a basis for getting fully computable two-sided bounds for the distance to the solution of the problem. The identities and the estimates derived from them can be used to estimate errors of approximations generated by various numerical methods. They are true both for solenoidal approximations as well as for those that satisfy the incompressibility condition only with a certain degree of accuracy. In addition, these identities and estimates permit one to compare exact solutions of problems with different data. Therefore, they open a way to evaluate errors of mathematical models, e.g., those that arise after changing (simplification) of the differential equation or due to replacing the incompressibility condition by weaker conditions.