Estimates of Deviations from Exact Solutions for Boundary-value Problems with Incompressibility Condition

Methods of estimating the difference between exact and approximate solutions are considered for boundary-value problems in spaces of solenoidal functions. The estimates obtained apply to any functions in the energy space of the respective problem, and their computation requires solving only finite-dimensional problems. In the paper, two different methods are considered: one involves variational formulations and duality theory, and in the other, estimates are obtained from the integral identities that define generalized solutions of the problems in question. It is shown that estimates of deviations from an exact solution must include an additional penalty term with a factor determined by the constant in the Ladyzhenskaya–Babuška–Brezzi condition. §