We present and analyse a hybridized discontinuous Galerkin method for incompressible flow problems using non-affine cells, proving that it preserves a key invariance property that illudes most methods, namely that any irrotational component of the prescribed force is exactly balanced by the pressure gradient and does not influence the velocity field. This invariance property can be preserved in the discrete problem if the incompressibility constraint is satisfied in a sufficiently strong sense. We derive sufficient conditions to guarantee discretely divergence-free functions are exactly divergence-free, and give examples of divergence-free finite elements on meshes containing triangular, quadrilateral, tetrahedral, or hexahedral cells generated by a (possibly non-affine) map from their respective reference cells. In the case of quadrilateral cells, we prove an optimal error estimate for the velocity field that does not depend on the pressure approximation. Our theoretical analysis is supported by numerical results.

翻译：本文提出并分析了一种针对非仿射单元不可压缩流问题的混合化间断伽辽金方法，证明该方法保持了大多数算法所缺失的关键不变性，即给定力的任何无旋分量被压力梯度精确平衡，且不影响速度场。若不可压缩性约束以足够强的方式满足，则该不变性可在离散问题中得以保持。我们推导了保证离散散度自由函数严格无散的充分条件，并给出了由（可能非仿射的）参考单元映射生成的含三角形、四边形、四面体或六面体网格上的无散有限元示例。针对四边形网格，我们证明了速度场的最优误差估计，且该估计不依赖于压力逼近。理论分析得到了数值结果的验证。