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：本文研究并分析了一种用于稳态流体系统数值逼近的hp-版本内罚间断Galerkin有限元方法，其计算网格由边界上的多面体单元构成。该方法基于间断Galerkin方法，并借鉴了[13]中引入的任意形状单元技术进行拓展。在此框架下，通过将迹不等式、Markov型不等式及H1/L2型逆不等式推广至任意单元形状，我们研究了稳态Stokes流体系统，实现了inf-sup条件的证明及hp-先验误差估计，并通过数值方法探讨了最优收敛速率。当几何形变仅发生在边界多面体单元的边与面上时，该方法恢复并整合了间断Galerkin方法处理流体问题的灵活性与优越性，同时结合了基于任意形状单元的hp-版本技术的高效性，无需从给定参考框架进行任何映射。