论文标题
在DE RHAM复合物中的顶点贴片中P-bobust局部重建的约束和不受约束的稳定离散最小化
Constrained and unconstrained stable discrete minimizations for p-robust local reconstructions in vertex patches in the de Rham complex
论文作者
论文摘要
我们在四面体的斑块上分析了约束和不受约束的最小化问题，该斑块具有与程度p的不连续的分段多项式数据共享的共同顶点。我们表明，在H1，H（curl）或H（DIV）空间中符合度p的分段多项式空间中的离散最小化器与这些（无限维）Sobolev空间中的最小化器中的最小化器一样好，直至独立于P的常数。这些结果在有限元方法的分析和设计中很有用，即用于设计稳定的本地通勤投影仪，并在先验分析和后验误差估计的背景下建立局部最佳/全球最佳等价。先前在文献中已经治疗了H1（DIV）中H1中无约束的最小化和最小化的最小化。除了改善H1和H（DIV）案例中的结果外，我们的主要贡献是对H（Curl）框架的处理。这使我们能够在单个设置中以三个空间维度覆盖整个DE RHAM图。
We analyze constrained and unconstrained minimization problems on patches of tetrahedra sharing a common vertex with discontinuous piecewise polynomial data of degree p. We show that the discrete minimizers in the spaces of piecewise polynomials of degree p conforming in the H1, H(curl), or H(div) spaces are as good as the minimizers in these entire (infinite-dimensional) Sobolev spaces, up to a constant that is independent of p. These results are useful in the analysis and design of finite element methods, namely for devising stable local commuting projectors and establishing local-best/global-best equivalences in a priori analysis and in the context of a posteriori error estimation. Unconstrained minimization in H1 and constrained minimization in H(div) have been previously treated in the literature. Along with improvement of the results in the H1 and H(div) cases, our key contribution is the treatment of the H(curl) framework. This enables us to cover the whole De Rham diagram in three space dimensions in a single setting.
