DAI Hai, PAN Wenfeng. A Spectral Element Method for Transmission Eigenvalue Problems of the Helmholtz Equation[J]. Applied Mathematics and Mechanics, 2018, 39(7): 833-840. doi: 10.21656/1000-0887.380327
Citation: DAI Hai, PAN Wenfeng. A Spectral Element Method for Transmission Eigenvalue Problems of the Helmholtz Equation[J]. Applied Mathematics and Mechanics, 2018, 39(7): 833-840. doi: 10.21656/1000-0887.380327

A Spectral Element Method for Transmission Eigenvalue Problems of the Helmholtz Equation

doi: 10.21656/1000-0887.380327
  • Received Date: 2017-12-18
  • Rev Recd Date: 2018-01-18
  • Publish Date: 2018-07-15
  • A Chebyshev spectral element method for the transmission eigenvalue problems of the Helmholtz equation was proposed, which combined the flexibility of the finite element method to deal with the boundary and region and the fast convergence of the spectral method. By means of the principle of weighted residuals, the basic theory and mathematical formulae of the Chebyshev spectral element method for transmission eigenvalue problems were obtained. The original problem was transformed into quadratic eigenvalue problems. Furthermore, several numerical examples were given to illustrate the effectiveness of the proposed method.
  • loading
  • [1]
    COLTOND, MONK P. The inverse scattering problem for time-hormonic acoustic waves in an inhomogeneous medium[J]. The Quarterly Journal of Mechanics and Applied Mathematics,1988,26(1): 323-350.
    [2]
    COLTON D, PAIVARINTA L, SYLVESTER J. The interior transmission problem[J]. Inverse Problems & Imaging,2017,1(1): 13-28.
    [3]
    CAKONI F, COLTON D, HADDARH. On the determination of Dirichlet or transmission eigenvalues from far field data[J]. Comptes Rendus Mathematique,2012,348(7): 379-383.
    [4]
    CAKONI F, KRESS R. A boundary integral equation method for the transmission eigenvalue problem[J]. Applicable Analysis,2016,96(1): 23-38.
    [5]
    COLTON D, MONK P, SUN J. Analytical and computational methods for transmission eigenvalues[J]. Inverse Problems,2010,26(26): 045011.
    [6]
    JI X, SUN J, TURNER T. Algorithm 922: a mixed finite element method for Helmholtz transmission eigenvalues[J]. ACM Transactions on Mathematical Software,2012,38(4): 1-8.
    [7]
    JI X, SUN J, XIE H. A multigrid method for Helmholtz transmission eigenvalue problems[J]. Journal of Scientific Computing,2014,60(2): 276-294.
    [8]
    SUN J. Iterative methods for transmission eigenvalues[J]. Society for Industrial and Applied Mathematics,2011,49(49): 1860-1874.
    [9]
    AN J, SHEN J. Spectral approximation to a transmission eigenvalue problem and its applications to an inverse problem[J]. Computers & Mathematics With Applications,2015,69(10): 1132-1143.
    [10]
    ORSZAG S A. Spectral methods for problems in complex geometries[J]. Journal of Computational Physics,1980,37(1): 70-92.
    [11]
    PATERA A T. A spectral element method for fluid dynamics: laminar flow in a channel expansion[J]. Journal of Computational Physics,1984,54(3): 468-488.
    [12]
    容志建, 许传炬. 基于张量乘积的快速谱元算法[J]. 数学研究, 2008,41(3): 264-271.(RONG Zhijian, XU Chuanju. Tensor product based fast spectral element solves[J]. Journal of Mathematical Study,2008,41(3): 264-271.(in Chinese))
    [13]
    林伟军. 弹性波传播模拟的Chebyshev谱元法[J]. 声学学报, 2007,32(6): 525-533.(LIN Weijun. A Chebyshev spectral element method for elastic wave modeling[J]. Acta Acustica,2007,32(6): 525-533.(in Chinese))
    [14]
    周欣, 李铁香. Helmholtz方程透射特征值问题的数值算法[J]. 应用数学进展, 2016,5(4): 683-694.(ZHOU Xin, LI Tiexiang. Numerical solution of transmission eigenvalue problems of Helmholtz equation[J]. Advances in Applied Mathematics,2016,5(4): 683-694. (in Chinese))
    [15]
    朱晓钢, 聂玉峰. 变系数分数阶对流扩散方程的一种算子矩阵方法[J]. 应用数学和力学, 2018,39(1): 104-112.(ZHU Xiaogang, NIE Yufeng. An operational matrix method for fractional advection-diffusion equations with variable coefficients[J]. Applied Mathematics and Mechanics,2018,39(1): 104-112.(in Chinese))
    [16]
    朱昌允, 秦国良, 徐忠. Chebyshev谱元方法结合并行算法求解三维区域的Helmholtz方程[J]. 应用力学学报, 2012,29(3): 247-251.(ZHU Changyun, QIN Guoliang, XU Zhong. Parallel Chebyshev spectral element method for Helmholtz equation in 3D domain[J]. Chinese Journal of Applied Mechanics,2012,29(3): 247-251.(in Chinese))
  • 加载中

Catalog

    通讯作者: 陈斌, bchen63@163.com
    • 1. 

      沈阳化工大学材料科学与工程学院 沈阳 110142

    1. 本站搜索
    2. 百度学术搜索
    3. 万方数据库搜索
    4. CNKI搜索

    Article Metrics

    Article views (1131) PDF downloads(616) Cited by()
    Proportional views
    Related

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return