笛卡尔网格上大波数时谐Maxwell方程棱边元方法的多项式保持恢复

中文摘要 AI 辅助

本文研究在笛卡尔网格上，针对大波数三维时谐Maxwell方程的最低阶第一类Nédélec棱边元方法（EEM）。分别针对棱边元解的旋度和解本身，提出了新的多项式保持恢复（PPR）算子。在$\kappa^3 h^2 C_{\mathrm{sol}}$足够小的条件下，证明了恢复的旋度和恢复的解均具有二阶超收敛估计，其中$\kappa$为波数，$h$为网格尺寸，$C_{\mathrm{sol}}$为与Maxwell解算子相关的稳定性常数。特别地，分析表明所提出的PPR过程无法缓解EEM固有的著名污染效应。为减少污染误差，我们进一步提出了一种新的连续内部罚棱边元方法（CIP-EEM），该方法引入了额外的法向跳跃罚项。结果表明，通过适当选择罚参数，新的CIP-EEM可将相位误差在$\kappa h$上提高两个阶次。数值实验验证了理论超收敛结果，并证明CIP-EEM能有效减少高频区域的污染误差。

英文摘要

This paper considers the lowest-order first type Nédélec edge element method (EEM) on Cartesian grids for the three-dimensional time-harmonic Maxwell equations with a large wave number. New polynomial preserving recovery (PPR) operators are proposed for the curl of the edge element solution and for the solution itself, respectively. Under the condition that $κ^3 h^2 C_{\mathrm{sol}}$ is sufficiently small, second-order superconvergence estimates are proved for both the recovered curl and the recovered solution, where $κ$ is the wave number, $h$ is the mesh size, and $C_{\mathrm{sol}}$ is a stability constant associated with the Maxwell solution operator. In particular, the analysis shows that the proposed PPR procedures cannot mitigate the well-known pollution effect inherent to the EEM. To reduce the pollution error, we further propose a new continuous interior penalty edge element method (CIP-EEM) that incorporates an additional normal-jump penalty term. It is shown that by appropriately choosing the penalty parameters, the new CIP-EEM can improve the phase error by two orders in $κh$. Numerical experiments are presented to confirm the theoretical superconvergence results and to demonstrate that the CIP-EEM can effectively reduce the pollution error in the high-frequency regime.

发表机构

• School of Mathematics, Nanjing University（南京大学数学系）

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