Preasymptotic Error Analysis of Higher Order FEM and CIP-FEM for Helmholtz Equation with High Wave Number
Preasymptotic Error Analysis of Higher Order FEM and CIP-FEM for Helmholtz Equation with High Wave Number
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DOI：
10.1137/140953125
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发表时间：
2014-01
期刊：
影响因子：
--
通讯作者：
Yu Du;Haijun Wu
中科院分区：
文献类型：
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作者：
Yu Du;Haijun Wu
本文对二维和三维Helmholtz方程的有限元法（FEM）和某些连续内点罚有限元法（CIP-FEM）进行了预渐近误差分析。给出了H^1 $和L^2 $两个误差估计，它们与波数k有显式的依赖关系.特别地，证明了如果k^{2 p +1}h^{2 p}$足够小，则这两种方法的污染误差在H^1 $-范数下有界为O（k^{2 p +1}h^{2 p}）$，这与已有的笛卡尔网格上的频散分析所得到的有限元法的相位误差相一致，其中h$为网格尺寸，$p$是近似空间的阶，是固定的。CIP有限元法扩展了经典的方法，在高阶（最高p$阶）法向导数的跳跃上增加了更多的惩罚项，有效地减小了高阶方法的污染误差。数值试验验证了理论结果，并说明了CIP-FEM在减少污染影响方面的巨大能力。
A preasymptotic error analysis of the finite element method (FEM) and some continuous interior penalty finite element method (CIP-FEM) for the Helmholtz equation in two and three dimensions is proposed. $H^1$- and $L^2$-error estimates with explicit dependence on the wave number $k$ are derived. In particular, it is shown that if $k^{2p+1}h^{2p}$ is sufficiently small, then the pollution errors of both methods in $H^1$-norm are bounded by $O(k^{2p+1}h^{2p})$, which coincides with the phase error of the FEM obtained by existent dispersion analyses on Cartesian grids, where $h$ is the mesh size, and $p$ is the order of the approximation space and is fixed. The CIP-FEM extends the classical one by adding more penalty terms on jumps of higher (up to $p$th order) normal derivatives in order to reduce efficiently the pollution errors of higher order methods. Numerical tests are provided to verify the theoretical findings and to illustrate the great capability of the CIP-FEM in reducing the pollution effect.