双调和问题的最优内罚不连续Galerkin方法

Zhaonan Dong, Lorenzo Mascotto
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引用次数: 2

摘要

对于具有齐次基本边界条件的双调和问题,我们证明了内罚不连续Galerkin方法的$hp$最优误差估计。我们在二维和三维中考虑张量积型网格,在二维中考虑三角形网格。分析中的一个重要组成部分是在给定网格上构造一个具有最优逼近性质的全局$H^2$分段多项式近似。讨论了二维和三维数学C^0$-IPDG的$hp$-最优性,以及二维Stokes问题的流形式。数值实验验证了理论预测,并揭示了$p$-次最优性存在于奇异基本边界条件下。
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hp-optimal interior penalty discontinuous Galerkin methods for the biharmonic problem
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.
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