This paper develops and analyses semi-discrete numerical method for two dimensional Vlasov-Stokes' system with periodic boundary condition. The method is based on coupling of semi-discrete discontinuous Galerkin method for the Vlasov equation with discontinuous Galerkin scheme for the stationary incompressible Stokes' equation. The proposed method is both mass and momentum conservative. Since it is difficult to establish non-negativity of the discrete local density, the generalized discrete Stokes' operator become non-coercive and indefinite and under smallness condition on the discretization parameter, optimal error estimates are established with help of a modified the Stokes' projection to deal with Stokes' part and with the help of a special projection to tackle the Vlasov part. Finally, numerical experiments based on the dG method combined with a splitting algorithm are performed.

翻译：本文针对具有周期边界条件的二维Vlasov-Stokes系统，提出并分析了半离散数值方法。该方法采用半离散间断伽辽金格式耦合求解Vlasov方程与稳态不可压Stokes方程。所提方法同时满足质量守恒和动量守恒。由于难以确保离散局部密度的非负性，广义离散Stokes算子失去强制性和正定性。在离散参数的微小性条件下，通过修正的Stokes投影处理Stokes部分，并借助特殊投影处理Vlasov部分，建立了最优误差估计。最后，基于dG方法与分裂算法相结合进行了数值实验。