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.

通过局部压力残差稳定作为经典 Galerkin/最小二乘 （GLS） 稳定方法的增强，为不可压缩流中的一类斯托克斯方程提出了一种新的基于局部评估的稳定有限元方法。我们专注于研究一种涉及混合切向速度和压力狄利克雷边界条件的非标准边界条件。出乎意料的是，与标准的无滑移速度狄利克雷边界条件形成鲜明对比的是，离散的 LBB inf-sup 稳定元和稳定方法（如经典的 GLS 方法）都无法确保收敛的有限元解，因为速度解可能非常弱，其梯度不是平方可积的。本文的主要目的是研究用于近似极弱速度解的新型稳定方法的误差估计;通过局部压力残差稳定，我们可以设法以合理的收敛阶数证明误差估计。给出了所提方法的性能和理论结果。