In this paper, we introduce an \(H({\text {div}})\) finite element method on polygonal and polyhedral meshes for solving the Stokes equations in the primary velocity–pressure formulation. An \(H({\text {div}})\) finite element on polygons or polyhedra is introduced to approximate the velocity so that the method is pressure robust and produces exact divergence free solutions, when the discontinuous \(P_k\) finite element is adopted for the pressure approximation. In addition, this method is stabilizer-free, in handling the \(H^1\) non-conformity. Optimal order error estimates are established for the method. Numerical tests are conducted to confirm the theory.

在本文中，我们引入了多边形和多面体网格上的\(H({\text {div}})\)有限元方法，用于求解初级速度-压力公式中的斯托克斯方程。引入多边形或多面体上的\({\text {div}})\)有限元来近似速度，以便该方法具有压力鲁棒性，并在不连续\(P_k\)有限时产生精确的无散解。采用压力近似元件。另外，该方法在处理\(H^1\)不合格时无需使用稳定剂。为该方法建立了最佳阶次误差估计。进行数值测试来证实该理论。