We generalise a hybridized discontinuous Galerkin method for incompressible flow problems to non-affine cells, showing that with a suitable element mapping the generalised method preserves a key invariance property that eludes most methods, namely that any irrotational component of the prescribed force is exactly balanced by the pressure gradient and does not affect 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 with 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 analysis is supported by numerical results.

翻译：我们将一种用于不可压缩流动问题的杂交间断伽辽金方法推广至非仿射单元，证明了在合适的单元映射下，该方法保留了一个大多数方法缺失的关键不变性质：即给定作用力中任何无旋分量被压力梯度精确平衡，且不影响速度场。若不可压缩约束以足够强的形式得到满足，则该不变性质可在离散问题中得以保留。我们推导了确保离散散度自由函数严格散度自由的充分条件，并给出了由参考单元经（可能非仿射的）映射生成的三角形、四边形、四面体或六面体网格上的散度自由有限元实例。针对四边形单元情形，我们证明了不依赖于压力近似的速度场最优误差估计。数值结果验证了我们的理论分析。