Unilateral problem for the Stokes equations: The well-posedness and finite element approximation

Abstract

We consider the stationary Stokes equations under a unilateral boundary condition of Signorini's type, which is one of artificial boundary conditions in flow problems. Well-posedness is discussed through its variational inequality formulation. We also consider the finite element approximation for a regularized penalty problem. The well-posedness, stability and error estimates of optimal order are established. The lack of a coupled Babuška and Brezzi's condition makes analysis difficult. We offer a new method of analysis. Particularly, our device to treat the pressure is novel and of some interest. Numerical examples are presented to validate our theoretical results.