A posteriori estimates for the stationary Stokes problem in exterior domains

Abstract

This paper is concerned with the analysis of the inf-sup condition arising in the stationary Stokes problem in exterior domains and applications to the derivation of computable bounds for the distance between the exact solution of the exterior Stokes problem and a certain approximation (which may be of a rather general form). In the first part, guaranteed bounds are deduced for the constant in the stability lemma associated with the exterior domain. These bounds depend only on known constants and the stability constant related to bounded domains that arise after suitable truncations of the unbounded domains. The lemma in question implies computable estimates of the distance to the set of divergence free fields defined in exterior domains. Such estimates are crucial for the derivation of computable majorants of the difference between the exact solution of the Stokes problem in exterior domains and an approximation from the admissible (energy) class of functions satisfying the Dirichlet boundary condition but not necessarily divergence free (solenoidal). Estimates of this type are often called a posteriori estimates of functional type. The constant in the stability lemma (or equivalently in the inf-sup or LBB condition) serves as a penalty factor at the term that controls violations of the divergence free condition. In the last part of the paper, similar estimates are deduced for the distance to the exact solution for nonconforming approximations, i.e., for those that may violate some continuity and boundary conditions. The case where the dimension of the domain equals $ 2$ requires a special consideration because the corresponding weighted spaces differ from those natural for the dimension $ 3$ (or larger). This special case is briefly discussed at the end of the paper where similar estimates are deduced for the distance to the exact solution of the exterior Stokes problem.