论文标题
Helmholtz方程的CIP-FEM的分散分析
Dispersion Analysis of CIP-FEM for Helmholtz Equation
论文作者
论文摘要
在数值上求解Helmholtz方程时，数值解的精度随着波数$ K $的增加而恶化，称为“污染效应”，这与数值分散剂引起的精确和数值溶液之间的相位差直接相关。在本文中，我们提出了连续内部惩罚有限元方法（CIP-FEM）的分散分析，并得出了$ p^{\ rm th} $ cip-fem tensor产品（cartesian）网格中的罚款参数的明确公式，从而从中降低了相位差异。 $ \ MATHCAL {o} \ big（k（kh）^{2p} \ big）$ to $ \ MATHCAL {O} \ big（k（kh）^{2p+2} \ big）$。广泛的数值测试表明，CIP-FE解决方案的污染误差也通过$ kh $中的两个订单降低，具有相同的惩罚参数。
When solving the Helmholtz equation numerically, the accuracy of numerical solution deteriorates as the wave number $k$ increases, known as `pollution effect' which is directly related to the phase difference between the exact and numerical solutions, caused by the numerical dispersion. In this paper, we propose a dispersion analysis for the continuous interior penalty finite element method (CIP-FEM) and derive an explicit formula of the penalty parameter for the $p^{\rm th}$ order CIP-FEM on tensor product (Cartesian) meshes, with which the phase difference is reduced from $\mathcal{O}\big(k(kh)^{2p}\big)$ to $\mathcal{O}\big(k(kh)^{2p+2}\big)$. Extensive numerical tests show that the pollution error of the CIP-FE solution is also reduced by two orders in $kh$ with the same penalty parameter.
