用于间断Galerkin方法完整离散$W^{1,\fty}$分析的界面格林函数框架

中文摘要 AI 辅助

过去二十年，间断Galerkin（DG）方法的逐点误差分析受到大量关注。现有分析已针对多种DG方法建立了$L^{\fty}$范数和间断$W^{1,\fty}$半范数的最优误差估计。然而，由于间断$W^{1,\fty}$半范数无法控制网格界面处的间断，针对间断近似的完整离散$W^{1,\fty}$理论仍未出现。本文针对凸多面体域上的DG方法，开发了一种用于完整离散$W^{1,\fty}$分析的界面格林函数框架。该框架引入了新的界面格林函数，用以表示间断近似在网格界面处的跳跃。其核心分析要素是针对这些格林函数的新局部能量估计，该估计通过泰勒展开论证得到，利用了相邻离散δ函数间的抵消作用。此估计与现有格林函数分析中的估计存在本质差异，可在不引入额外对数因子的情况下推导界面估计。

英文摘要

Pointwise error analysis of discontinuous Galerkin (DG) methods for the Poisson equation has received considerable attention during the past two decades. However, on convex polyhedral domains, existing analyses can only establish optimal error estimates in the broken $W^{1,\infty}$ seminorm for several DG methods. Since the broken $W^{1,\infty}$ seminorm does not control discontinuities across mesh interfaces, a complete discrete $W^{1,\infty}$ theory for discontinuous approximations on convex polyhedral domains has remained unavailable. In this paper, we develop an interface Green's function framework for the complete discrete $W^{1,\infty}$ analysis of DG methods on convex polyhedral domains. The proposed framework introduces new interface Green's functions that represent jumps of discontinuous approximations across mesh interfaces. Its central analytical ingredient is a new local energy estimate for these Green's functions, obtained by exploiting a cancellation between neighboring discrete delta functions. This estimate differs fundamentally from existing Green's function estimates and enables us to derive maximum-norm estimate for interface jumps without introducing additional logarithmic factors.

发表机构

• Guangzhou University（广州大学）
• City University of Hong Kong（香港城市大学）

机构由 AI 辅助整理，请以论文原文为准。