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：我们证明了在齐次本质边界条件下，内罚间断伽辽金方法（IPDG）用于双调和问题的$hp$-最优误差估计。分析中考虑了二维和三维的张量积类型网格，以及二维的三角形网格。分析的一个关键要素是在给定网格上构造具有$hp$-最优逼近性质的全局$H^2$分片多项式逼近子。本文还讨论了二维和三维$\mathcal C^0$-IPDG的$hp$-最优性，以及二维Stokes问题的流函数形式。数值实验验证了理论预测，并揭示了在奇异本质边界条件下会出现$p$-次优性。