On a Priori Error Analysis of Discontinuous Galerkin Method for the Vlasov-Nonstationary Stokes System

Abstract

In the first part of this paper, the uniqueness of a strong solution is established for the Vlasov-unsteady Stokes problem in 3D. The second part deals with a semi discrete scheme, which is derived as a result of spatial discretization of the coupled system of Vlasov and Stokes equations for the 2D problem by discontinuous Galerkin methods, while keeping temporal variable continuous. The proposed semi-discrete scheme preserves both mass and momentum conservation properties. Based on the orthogonal $L^2$ as well as the Stokes projections, error estimates in the case of smooth compactly supported initial data are derived by employing a variant of nonlinear Grönwall’s lemma in a crucial way. Moreover, the generalization of error estimates to 3D problem is also briefly discussed. Finally, using a time splitting algorithm as the phase space is four dimensional, some numerical experiments are conducted, whose results confirm our theoretical findings.