We introduce a new discretization based on the Trefftz-DG method for solving the Stokes equations. Discrete solutions of a corresponding method fulfill the Stokes equation pointwise within each element and yield element-wise divergence-free solutions. Compared to standard DG methods, a strong reduction of the degrees of freedom is achieved, especially for higher order polynomial degrees. In addition, in contrast to many other Trefftz-DG methods, our approach allows to easily incorporate inhomogeneous right hand sides (driving forces) by using the concept of the embedded Trefftz-DG method. On top of a detailed a priori error analysis, we further compare our approach to standard discontinuous Galerkin Stokes discretizations and present numerical examples.

翻译：我们提出了一种基于Trefftz-DG方法的新型离散化方案，用于求解斯托克斯方程。相应方法的离散解在每个单元内部逐点满足斯托克斯方程，并得到单元级别的无散度解。与标准DG方法相比，该方案显著减少了自由度数量，尤其在采用高阶多项式次数时更为突出。此外，与许多其他Trefftz-DG方法不同，我们的方法通过引入嵌入式Trefftz-DG概念，能够轻松处理非齐次右端项（驱动力）。基于详细的先验误差分析，我们进一步将本方法与标准间断伽辽金斯托克斯离散化进行了对比，并给出了数值算例。