Estimates of the Distance to the Set of Solenoidal Vector Fields and Applications to A Posteriori Error Control

Abstract

AbstractThe paper is concerned with computable estimates of the distance between a vector-valued function in the Sobolev space$W^{1,\gamma }(\Omega ,\mathbb {R}^d)$(where${\gamma \in (1,+\infty )}$and Ω is a bounded Lipschitz domain in ℝd) and the subspace${S^{1,\gamma }(\Omega ,\mathbb {R}^d)}$containing all divergence-free (solenoidal) vector functions. Derivation of these estimates is closely related to the stability theorem that establishes existence of a bounded operator inverse to the operator${\operatorname{div}}$. The constant in the respective stability inequality arises in the estimates of the distance to the set${S^{1,\gamma }(\Omega ,\mathbb {R}^d)}$. In general, it is difficult to find a guaranteed and realistic upper bound of this global constant. We suggest a way to circumvent this difficulty by using weak (integral mean) solenoidality conditions and localized versions of the stability theorem. They are derived for the case where Ω is represented as a union of simple subdomains (overlapping or non-overlapping), for which estimates of the corresponding stability constants are known. These new versions of the stability theorem imply estimates of the distance to${S^{1,\gamma }(\Omega ,\mathbb {R}^d)}$that involve only local constants associated with subdomains. Finally, the estimates are used for deriving fully computable a posteriori estimates for problems in the theory of incompressible viscous fluids.