In a polygon |$\varOmega \subset \mathbb{R}^2$| we consider mixed |$hp$|-discontinuous Galerkin approximations of the stationary, incompressible Navier–Stokes equations, subject to no-slip boundary conditions. We use geometrically corner-refined meshes and |$hp$| spaces with linearly increasing polynomial degrees. Based on recent results on analytic regularity of velocity field and pressure of Leray solutions in |$\varOmega$|⁠, we prove exponential rates of convergence of the mixed |$hp$|-discontinuous Galerkin finite element method, with respect to the number of degrees of freedom, for small data which is piecewise analytic.

中文翻译：

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ℝ ^{2 中}不可压缩 Navier-Stokes 方程的混合 hp-DGFEM 指数收敛

在多边形中|$\varOmega \subset \mathbb{R}^2$| 我们考虑混合|$hp$| - 静止的、不可压缩的 Navier-Stokes 方程的不连续 Galerkin 近似，受无滑移边界条件的约束。我们使用几何角细化网格和|$hp$| 多项式次数线性增加的空间。基于最近关于|$\varOmega$|⁠中 Leray 解的速度场和压力的解析规律的结果，我们证明了混合|$hp$| 的指数收敛速度。- 不连续伽辽金有限元方法，关于自由度数，对于分段分析的小数据。