标题： Sobolev类似空间中的拉格朗日有限元$3/2$ 显示英文标题

标题： Lagrangian finite elements in Sobolev-like spaces of order $3/2$

摘要： 本文引入了一个阶数为$3/2$的类似Sobolev空间，记为$\widehat{H}^{3/2}$，用于拉格朗日有限元，特别是$C^0$元素。其动机来自于当前演化曲面有限元方法（ESFEM）稳定性分析的局限性，该分析仅依赖于能量估计框架。为了建立基于偏微分方程（PDE）的ESFEM分析框架，我们遇到了一个基本的正则性不匹配问题：ESFEM采用的是$C^0$元素，而PDE正则性理论要求解具有$H^{3/2}$正则性。为克服这一困难，我们首先研究连续$H^{3/2}$空间的性质，然后引入一个狄利克雷提升和Scott-Zhang类型的插值算子，以连接到离散的$\widehat{H}^{3/2}$空间。 我们新的$\widehat{H}^{3/2}$空间被证明与椭圆PDE正则性理论、迹不等式和逆不等式相容。值得注意的是，我们将ESFEM中的关键域变形估计扩展到了$\widehat{H}^{3/2}$设置中。$\widehat{H}^{3/2}$理论为建立基于PDE的ESFEM收敛分析框架提供了基础。 显示英文摘要

摘要： This paper introduces a Sobolev-like space of order $3/2$, denoted as $\widehat{H}^{3/2}$, for Lagrangian finite elements, especially for $C^0$ elements. It is motivated by the limitations of current stability analysis of the evolving surface finite element method (ESFEM), which relies exclusively on an energy estimate framework. To establish a PDE-based analysis framework for ESFEM, we encounter a fundamental regularity mismatch: the ESFEM adopts the $C^0$ elements, while the PDE regularity theory requires $H^{3/2}$ regularity for solutions. To overcome this difficulty, we first examine the properties of the continuous $H^{3/2}$ space, then introduce a Dirichlet lift and Scott-Zhang type interpolation operators to bridge to the discrete $\widehat{H}^{3/2}$ space. Our new $\widehat{H}^{3/2}$ space is shown to be compatible with the elliptic PDE regularity theory, the trace inequality, and the inverse inequality. Notably, we extend the critical domain deformation estimate in ESFEM to the $\widehat{H}^{3/2}$ setting. The $\widehat{H}^{3/2}$ theory provides a foundation for establishing a PDE-based convergence analysis framework of ESFEM.