The sub-Cauchy-Stokes problem: Solvability issues and Lagrange multiplier methods with artificial boundary conditions

In contrast to the conventional Cauchy-Stokes problems, in which the velocity and the stress force data are given on the accessible boundary, in the present paper, we reduce the accessible boundary data information and we consider a problem which deals only with shear stress data. We refer to this problem as a sub-Cauchy-Stokes problem. This problem is ill-posed because of severe instability and even uniqueness is unknown. We first address the uniqueness issues associated with this problem. Resorting to the domain decomposition techniques together with the duplication process of Vogelius (Kohn and Vogelius, 1985), we propose new Lagrange multiplier methods to solve the sub-Cauchy-Stokes problem. These methods consist in recasting the problem in terms of interfacial equations, by equalizing two solutions of the sub-Cauchy-Stokes problem using matching conditions defined on the inaccessible boundary. The matching is based on second order conditions and the types of the interfacial equations depend on the equations used to match the values of the unknowns on the inaccessible boundary. The interfacial problems are then solved by iterative procedures in which coefficients can be optimized to improve convergence rates. A complete analysis of the methods is presented, and intensive numerical results illustrate the effectiveness and the performance of the proposed approaches.