Linear continuous interior penalty finite element method for Helmholtz equation With High Wave Number: One-Dimensional Analysis
Linear continuous interior penalty finite element method for Helmholtz equation With High Wave Number: One-Dimensional Analysis
复制标题
高波数亥姆霍兹方程的线性连续内罚有限元法：一维分析
DOI：
10.1002/num.22054
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发表时间：
2016
影响因子：
3.9
通讯作者：
Zhu Lingxue
中科院分区：
文献类型：
--
作者：
Burman Erik;Wu Haijun;Zhu Lingxue
讨论了Helmholtz方程的连续内罚有限元解的性质。将CIP有限元法的分段线性逼近形式应用于一维（1D）模型问题。我们首先显示离散的适定性和收敛结果，使用的虚部的稳定算子，复杂的亥姆霍兹方程。然后我们考虑一个具有真实的值罚参数的方法，并证明了离散解在-范数下的误差估计，即最佳逼近误差加上一个污染项的总和，该污染项是相位差的阶数。证明了通过适当选择惩罚参数可以消除污染效应。作为这种分析的结果，在整个收敛范围内获得了对误差行为的透彻和严格的理解。数值结果表明，尖锐的误差估计和突出的离散解的行为的一些现象。特别是，我们给出了数值证据，证明在一维情况下获得的最优惩罚参数也适用于二维笛卡尔网格上的CIP‐FEM。© 2016 Wiley Periodicals，Inc. Numer Methods Partial Differential Eq 32：1378-1410，2016
This article addresses the properties of continuous interior penalty (CIP) finite element solutions for the Helmholtz equation. The ‐version of the CIP finite element method with piecewise linear approximation is applied to a one‐dimensional (1D) model problem. We first show discrete well posedness and convergence results, using the imaginary part of the stabilization operator, for the complex Helmholtz equation. Then we consider a method with real valued penalty parameter and prove an error estimate of the discrete solution in the ‐norm, as the sum of best approximation error plus a pollution term that is the order of the phase difference. It is proved that the pollution effect can be eliminated by selecting the penalty parameter appropriately. As a result of this analysis, thorough and rigorous understanding of the error behavior throughout the range of convergence is gained. Numerical results are presented that show sharpness of the error estimates and highlight some phenomena of the discrete solution behavior. In particular, we give numerical evidence that the optimal penalty parameter obtained in the 1D case also works very well for the CIP‐FEM on two‐dimensional Cartesian grids.© 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1378–1410, 2016