In this paper we provide key estimates used in the stability and error analysis of discontinuous Galerkin finite element methods (DGFEMs) on domains with curved boundaries. In particular, we review trace estimates, inverse estimates, discrete Poincaré–Friedrichs' inequalities, and optimal interpolation estimates in noninteger Hilbert–Sobolev norms, that are well known in the case of polytopal domains. We also prove curvature bounds for curved simplices, which does not seem to be present in the existing literature, even in the polytopal setting, since polytopal domains have piecewise zero curvature. We demonstrate the value of these estimates, by analyzing the IPDG method for the Poisson problem, introduced by Douglas and Dupont, and by analyzing a variant of the hp‐DGFEM for the biharmonic problem introduced by Mozolevski and Süli. In both cases we prove stability estimates and optimal a priori error estimates. Numerical results are provided, validating the proven error estimates.

中文翻译：

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弯曲域上的有限元理论及其在不连续Galerkin有限元方法中的应用

在本文中，我们提供了在具有弯曲边界的域上的不连续Galerkin有限元方法（DGFEM）的稳定性和误差分析中使用的关键估计。特别是，我们回顾了多点域中众所周知的非整数Hilbert-Sobolev规范中的迹线估计，逆估计，离散Poincaré-Friedrichs不等式和最佳插值估计。我们还证明了弯曲单曲面的曲率边界，即使在多曲面环境中，在现有文献中似乎也不存在，因为多曲面域的分段曲率为零。我们通过分析Douglas和Dupont提出的针对Poisson问题的IPDG方法，以及分析hp的变体，来证明这些估计的价值‐DGFEM用于解决Mozolevski和Süli提出的双谐波问题。在这两种情况下，我们都证明了稳定性估计和最佳先验误差估计。提供了数值结果，验证了经验证的误差估计。