论文标题
$ hp $ - 最佳的内部罚款不连续的Galerkin方法
$hp$-optimal interior penalty discontinuous Galerkin methods for the biharmonic problem
论文作者
论文摘要
我们证明了$ hp $ - 最佳的错误估计，用于内部罚款不连续的盖尔金方法（IPDG），用于具有均质基本边界条件的Biharmonic问题。我们考虑在两个和三个维度上的张量产品类型网格，以及在两个维度上的三角形网格。分析中的重要成分是在给定的网格上构建具有$ hp $ - 最佳近似属性的全局$ h^2 $分段多项式近似值。 $ hp $ - 最佳性在两个维度和三个维度上也以$ \ Mathcal C^0 $ -IPDG进行了讨论，以及在两个维度上的流式公式。数值实验验证了理论预测，并表明$ p $ - 贝映是在存在奇异基本边界条件下发生的。
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.
