|
|
任意凸四边形区域上二阶变系数椭圆边值问题有效的谱Galerkin逼近
|
Abstract:
本文提出了在任意凸四边形区域上二阶变系数椭圆边值问题的一种有效谱Galerkin逼近。 首先，通过双线性等参变换和坐标变换将任意四边形区域转换到D？= [？1, 1]2，并建立其在D？ 的弱形式及相应的离散格式。 其次，我们证明了弱解的存在唯一性。 另外，利用Legendre 正交多项式构 造了逼近空间中一组有效的基函数，推导出离散格式的矩阵形式。 最后通过数值实验，验证了 谱Galerkin逼近任意凸四边形区域上二阶变系数椭圆边值问题的谱收敛。
In this paper, an efficient spectral Galerkin approximation for second-order elliptic boundary value problems with variable coefficients on an arbitrary convex quadrilat-eral region is proposed. Firstly, any quadrilateral region is converted to D？= [？1, 1]2 by bilinear isoparametric transformation and coordinate transformation, and its weak form and corresponding discrete format on D？ are established. Secondly, we prove the existence and uniqueness of the weak solution. In addition, the Legendre orthogonal polynomial is used to construct a set of effective basis functions in the approximation space, and the matrix form of the discrete scheme is derived. Finally, the spectral convergence of spectral Galerkin approximation to the second-order elliptic boundary value problem with variable coefficients on arbitrary convex quadrilateral region is verified by numerical experiments.
| [1] | Boffi, D. (2010) Finite Element Approximation of Eigenvalue Problems. Acta Numerica, 19, 1-120. https://doi.org/10.1017/S0962492910000012 |
| [2] | Xu, J. and Zhou, A. (2002) Local and Parallel Finite Element Algorithms for Eigenvalue Problems. Acta Mathematicae Applicatae Sinica, 18, 185-200. https://doi.org/10.1007/s102550200018 |
| [3] | Hu, J., Huang, Y. and Shen, H. (2004) The Lower Approximation of Eigenvalue by Lumped Mass Finite Element Method. Computational Mathematics (English Edition), 22, 545-556. |
| [4] | Liu, F. and Shen, J. (2015) Stabilized Semi-Implicit Spectral Deferred Correction Methods for Allen-Cahn and Cahn-Hilliard Equations. Mathematical Methods in the Applied Sciences, 38, 4564-4575. https://doi.org/10.1002/mma.2869 |
| [5] | Colton, D., P¨aiv¨arinta, L. and Sylvester, J. (2007) The Interior Transmission Problem. Inverse Problems & Imaging, 1, 13-28. https://doi.org/10.3934/ipi.2007.1.13 |
| [6] | 陈婷婷. 强异向性椭圆方程的紧致有限差分法研究[D]: [硕士学位论文]. 哈尔滨: 哈尔滨工业大学, 2021. https://doi.org/10.27061/d.cnki.ghgdu.2021.001684 |
| [7] | 闵涛, 高青青. 求解二阶椭圆型方程Dirichlet问题的几种算法[J]. 科技通报, 2017, 33(12): 44-49. https://doi.org/10.13774/j.cnki.kjtb.2017.12.009 |
| [8] | 林府标, 杨一都. 有限元二网格离散方案EQrot元特征值下逼近准确特征值[J]. 贵州师范大学学报(自然科学版), 2008(2): 68-74. https://doi.org/10.16614/j.cnki.issn1004-5570.2008.02.001 |
| [9] | Sun, J. (2011) Iterative Methods for Transmission Eigenvalues. SIAM Journal on Numerical Analysis, 49, 1860-1874. https://doi.org/10.1137/100785478 |
| [10] | Perugia, I. and Sch¨otzau, D. (2002) An hp-Analysis of the Local Discontinuous Galerkin Method for Diffusion Problems. Journal of Scientific Computing, 17, 561-571. https://doi.org/10.1023/A:1015118613130 |
| [11] | Cakoni, F., Colton, D., Monk, P., et al. (2010) The Inverse Electromagnetic Scattering Problem for Anisotropic Media. Inverse Problems, 26, Article 074004. https://doi.org/10.1088/0266-5611/26/7/074004 |
| [12] | Guo, B.Y. and Jia, L.H. (2010) Spectral Method on Quadrilaterals. Mathematics of Compu- tation, 79, 2237-2264. https://doi.org/10.1090/S0025-5718-10-02329-X |
| [13] | Guo, B.Y. and Wang, L.L. (2007) Error Analysis of Spectral Method on a Triangle. Advances in Computational Mathematics, 26, 473-496. https://doi.org/10.1007/s10444-005-7471-8 |
| [14] | Shen, J. and Tang, T. (2006) Spectral and High-Order Methods with Applications. Science Press, Beijing. |
| [15] | Shen, J., Tang, T. and Wang, L.L. (2011) Spectral Methods: Algorithms, Analysis and Appli- cations. Springer, Science and Business Media. https://doi.org/10.1007/978-3-540-71041-7 |
| [16] | An, J. and Zhang, Z.M. (2018) An Efficient Spectral-Galerkin Approximation and Error Anal-ysis for Maxwell Transmission Eigenvalue Problems in Spherical Geometries. Journal of Sci- entific Computing, 75, 157-181. https://doi.org/10.1007/s10915-017-0528-2 |
| [17] | 郑继会. 任意凸四边形区域上二阶椭圆特征值问题基于高阶多项式逼近的一种数值方法[J]. 应 用数学进展, 2021, 10(12): 4201-4208. https://doi.org/10.12677/AAM.2021.1012446 |
