We prove $hp$-optimal error estimates for interior penalty discontinuous Galerkin methods (IPDG) for the biharmonic problem with homogeneous essential boundary conditions. We consider tensor product-type meshes in two and three dimensions, and triangular meshes in two dimensions. An essential ingredient in the analysis is the construction of a global $H^2$ piecewise polynomial approximants with $hp$-optimal approximation properties over the given meshes. The $hp$-optimality is also discussed for $\mathcal C^0$-IPDG in two and three dimensions, and the stream formulation of the Stokes problem in two dimensions. Numerical experiments validate the theoretical predictions and reveal that $p$-suboptimality occurs in presence of singular essential boundary conditions.

翻译：我们证明对内刑不连续加勒金方法(IPDG)中具有相同基本边界条件的双声调问题(IPDG)的美元-最佳误差估算值。我们从两个和三个层面考虑高产品型模贝,从两个层面考虑三角体模贝。分析中的一个基本成分是构建一个全球的$H$2,在给定的中间部分上带有$hp$-最佳近似属性的片段多角度多角度药剂。在两个和三个层面也讨论了$mathc%0$-IPDG的美元-最佳性能,在两个层面考虑斯托克斯问题流的配方。数字实验证实了理论预测,并揭示了在单一基本边界条件下存在美元-次优性。