In this paper, we consider a new nonlocal approximation to the linear Stokes system with periodic boundary conditions in two and three dimensional spaces . A relaxation term is added to the equation of nonlocal divergence free equation, which is reminiscent to the relaxation of local Stokes equation with small artificial compressibility. Our analysis shows that the well-posedness of the nonlocal system can be established under some mild assumptions on the kernel of nonlocal interactions. Furthermore, the new nonlocal system converges to the conventional, local Stokes system in second order as the horizon parameter of the nonlocal interaction goes to zero. The study provides more theoretical understanding to some numerical methods, such as smoothed particle hydrodynamics, for simulating incompressible viscous flows.

翻译：本文研究了两维和三维空间中具有周期边界条件的线性斯托克斯系统的一种新型非局部逼近方法。在非局部散度自由方程中添加了一个松弛项，这与局部斯托克斯方程通过小人工可压缩性进行松弛的方法类似。分析表明，在非局部相互作用核的某些温和假设下，该非局部系统的适定性得以建立。此外，当非局部相互作用的总域参数趋近于零时，新型非局部系统以二阶精度收敛至传统的局部斯托克斯系统。该研究为模拟不可压缩黏性流动的数值方法（如光滑粒子流体动力学）提供了更深入的理论理解。