$W^{s,p}$-逼近性质研究：椭圆投影在多项式空间上的应用，及其在混合高阶离散化勒雷-洛恩斯问题误差分析中的应用
$W^{s,p}$-approximation properties of elliptic projectors on polynomial spaces, with application to the error analysis of a Hybrid High-Order discretisation of Leray-Lions problems
AI总结：
本文研究了椭圆投影在局部多项式空间上的$W^{s,p}$逼近性质，通过经典Dupont-Scott理论和两个新引理，推导出最优逼近估计，并应用于混合高阶离散化勒雷-洛恩斯问题的误差分析。
摘要：
本文证明了在局部多项式空间上，椭圆投影的最优$W^{s,p}$逼近估计（$p\in[1,+\infty]$）。证明基于经典的Dupont-Scott逼近理论以及两个新的抽象引理：一个有界投影的逼近结果，以及一个$L^2$正交投影在多项式子空间上的$L^p$有界性结果。$W^{s,p}$逼近结果对基于局部多项式空间的标准或多面体数值方法具有普遍适用性。作为示例，我们利用这些$W^{s,p}$估计，推导出混合高阶离散化勒雷-洛恩斯椭圆问题的新型误差估计，其弱形式通常在$W^{1,p}(Ω)$中定义（$p\in(1,+\infty)$）。此类问题出现在冰川运动、不可压缩湍流流动和机翼设计建模中。设$h$为网格尺寸，我们证明在$W^{1,p}$-类离散范数下的逼近误差尺度为当$p\ge 2$时为$h^{\frac{k+1}{p-1}}$，当$p<2$时为$h^{(k+1)(p-1)}$。
英文摘要：
In this work we prove optimal $W^{s,p}$-approximation estimates (with $p\in[1,+\infty]$) for elliptic projectors on local polynomial spaces. The proof hinges on the classical Dupont--Scott approximation theory together with two novel abstract lemmas: An approximation result for bounded projectors, and an $L^p$-boundedness result for $L^2$-orthogonal projectors on polynomial subspaces. The $W^{s,p}$-approximation results have general applicability to (standard or polytopal) numerical methods based on local polynomial spaces. As an illustration, we use these $W^{s,p}$-estimates to derive novel error estimates for a Hybrid High-Order discretization of Leray--Lions elliptic problems whose weak formulation is classically set in $W^{1,p}(Ω)$ for some $p\in(1,+\infty)$. This kind of problems appears, e.g., in the modelling of glacier motion, of incompressible turbulent flows, and in airfoil design. Denoting by $h$ the meshsize, we prove that the approximation error measured in a $W^{1,p}$-like discrete norm scales as $h^{\frac{k+1}{p-1}}$ when $p\ge 2$ and as $h^{(k+1)(p-1)}$ when $p<2$.
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