A pressure-residual augmented GLS stabilized method for a type of Stokes equations with nonstandard boundary conditions

With local pressure-residual stabilizations as an augmentation to the classical Galerkin/least-squares (GLS) stabilized method, a new locally evaluated residual-based stabilized finite element method is proposed for a type of Stokes equations from the incompressible flows. We focus on the study of a type of nonstandard boundary conditions involving the mixed tangential velocity and pressure Dirichlet boundary conditions. Unexpectedly, in sharp contrast to the standard no-slip velocity Dirichlet boundary condition, neither the discrete LBB inf-sup stable elements nor the stabilized methods such as the classical GLS method could certainly ensure a convergent finite element solution, because the velocity solution could be very weak with its gradient not being square integrable. The main purpose of this paper is to study the error estimates of the new stabilized method for approximating the very weak velocity solution; with the local pressure-residual stabilizations, we can manage to prove the error estimates with a reasonable convergence order. Numerical results are provided to illustrate the performance and the theoretical results of the proposed method.