Locally Stabilized Finite Element Method for Stokes Problem with Nonlinear Slip Boundary Conditions

Abstract

Based on the low-order conforming finite element subspace $(V_h,M_h)$ such as the $P_1$-$P_0$ triangle element or the $Q_1$-$P_0$ quadrilateral element, the locally stabilized finite element method for the Stokes problem with nonlinear slip boundary conditions is investigated in this paper. For this class of nonlinear slip boundary conditions including the subdifferential property, the weak variational formulation associated with the Stokes problem is an variational inequality. Since $(V_h,M_h)$ does not satisfy the discrete inf-sup conditions, a macroelement condition is introduced for constructing the locally stabilized formulation such that the stability of $(V_h,M_h)$ is established. Under these conditions, we obtain the $H^1$ and $L^2$ error estimates for the numerical solutions.