In the present work, we examine and analyze an hp-version interior penalty discontinuous Galerkin finite element method for the numerical approximation of a steady fluid system on computational meshes consisting of polytopic elements on the boundary. This approach is based on the discontinuous Galerkin method, enriched by arbitrarily shaped elements techniques as has been introduced in [13]. In this framework, and employing extensions of trace, Markov-type, and H1/L2-type inverse estimates to arbitrary element shapes, we examine a stationary Stokes fluid system enabling the proof of the inf/sup condition and the hp- a priori error estimates, while we investigate the optimal convergence rates numerically. This approach recovers and integrates the flexibility and superiority of the discontinuous Galerkin methods for fluids whenever geometrical deformations are taking place by degenerating the edges, facets, of the polytopic elements only on the boundary, combined with the efficiency of the hp-version techniques based on arbitrarily shaped elements without requiring any mapping from a given reference frame.

翻译：本文研究并分析了在计算网格由边界多面体单元组成的稳态流体系统数值逼近中，一种hp-版本内部惩罚间断伽辽金有限元方法。该方法基于间断伽辽金方法，并采用[13]中引入的任意形状单元技术进行丰富。在此框架下，利用迹不等式、Markov型不等式以及H1/L2型逆估计在任意单元形状上的推广，我们研究了稳态Stokes流体系统，从而证明了inf-sup条件和hp-先验误差估计，同时通过数值实验探讨了最优收敛速度。该方法通过仅在边界上退化多面体单元的边或面来处理几何变形，结合基于任意形状单元的hp-版本技术的效率（无需从给定参考框架进行任何映射），恢复并融合了间断伽辽金方法在流体问题中的灵活性与优越性。