二維低波速Helmholtz方程的異質(zhì)多尺度-內(nèi)部懲罰間斷有限元方法

余 濤，李今欣

二維低波速Helmholtz方程的異質(zhì)多尺度-內(nèi)部懲罰間斷有限元方法

*余 濤，李今欣

（井岡山大學(xué)數(shù)理學(xué)院，江西，吉安 343009）

將異質(zhì)多尺度方法和內(nèi)部懲罰間斷有限元方法相結(jié)合，構(gòu)造了求解二維低波速Helmholtz方程的異質(zhì)多尺度-內(nèi)部懲罰間斷有限元方法，并在局部周期條件下給出了算法的最佳誤差估計(jì)。

Helmholtz方程；低波速；異質(zhì)多尺度方法；內(nèi)部懲罰間斷有限元方法；先驗(yàn)誤差估計(jì)

0 引言

考慮具有齊次Dirichlet邊界條件的Helmholtz方程

異質(zhì)多尺度方法[4-5]是求解多尺度問題的一個有效通用框架，其由兩部分組成：選擇合適的宏觀求解器和通過求解局部微觀問題估計(jì)缺失的宏觀系數(shù)。Ciarler和Stohrer采用標(biāo)準(zhǔn)有限元作為宏觀求解器構(gòu)造了求解高震蕩介質(zhì)中Helmholtz方程的異質(zhì)多尺度有限元方法[6]。Abdulle將異質(zhì)多尺度方法和間斷有限元方法相結(jié)合提出了求解純擴(kuò)散問題的算法[7-8]?，F(xiàn)將在異質(zhì)多尺度方法框架下采用內(nèi)部懲罰間斷有限元方法作為宏觀求解器，構(gòu)造求解二維低波速Helmholtz方程的異質(zhì)多尺度-內(nèi)部懲罰間斷有限元方法，并得到最佳的先驗(yàn)誤差估計(jì)。

1 異質(zhì)多尺度-內(nèi)部懲罰間斷有限元方法

與

其中

與

和

和

其中

與

2 誤差估計(jì)

在推導(dǎo)模型誤差時，先引入下面的引理。

先驗(yàn)誤差估計(jì)結(jié)果如下：

根據(jù)內(nèi)部懲罰間斷有限元方法的誤差估計(jì)[10-12]，可知宏觀誤差

又根據(jù)標(biāo)準(zhǔn)數(shù)值積分誤差分析[13]，結(jié)合數(shù)值積分精度（1.8）式易知

由文獻(xiàn)[7-8]中的結(jié)論

可知模型誤差

由（2.2）～（2.5）式，有

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HETEROGENEOUS MULTISCALE – INTERIOR PENALTY DISCONTINUOUS GALERKIN METHOD FOR TWO-DIMENSIONAL HELMHOLTZ EQUATION WITH LOW WAVE NUMBER

*YU Tao, LI Jin-xin

（School of Mathematics and Physics, Jinggangshan University, Ji’an, Jiangxi 343009, China）

By combining the heterogeneous multiscale method (HMM) with the interior penalty discontinuous Galerkin method (IPDG), the heterogenous multiscale-interior penalty discontinuous Galerkin method (HMM-IPDG) is constructed for two-dimensional Helmholtz equation with low wave number. The optimal error estimation of the algorithm is given under the condition of micro period.

Helmholtz equation; low wave number; heterogenous multiscale method; interior penalty discontinuous Galerkin method; priori error estimate

1674-8085（2023）05-0001-05

O242.1

A

10.3969/j.issn.1674-8085.2023.05.001

2022-09-29；

2022-11-25

江西省教育廳科技計(jì)劃項(xiàng)目(GJJ211027)；井岡山大學(xué)博士科研啟動項(xiàng)目(JZB1921)

*余 濤(1983-），男，江西萬安人，副教授，博士，主要從事多尺度建模研究(E-mail:yutao@jgsu.edu.cn).