Symmetric Homotopy Method for Discretized Elliptic Equations with Cubic and Quintic Nonlinearities
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- 作者：Xuping Zhang ; Jintao Zhang ; Bo Yu
- 关键词：Semilinear elliptic equation ; Boundary value problem ; Eigenfunction expansion ; Homotopy continuation ; Polynomial system
- 刊名：Journal of Scientific Computing
- 出版年：2017
- 出版时间：March 2017
- 年：2017
- 卷：70
- 期：3
- 页码：1316-1335
- 全文大小：
- 刊物类别：Mathematics and Statistics
- 刊物主题：Algorithms; Computational Mathematics and Numerical Analysis; Appl.Mathematics/Computational Methods of Engineering; Theoretical, Mathematical and Computational Physics;
- 出版者：Springer US
- ISSN：1573-7691
- 卷排序：70
文摘
Symmetry is analyzed in the solution set of the polynomial system resulted from the eigenfunction expansion discretization of semilinear elliptic equation with polynomial nonlinearity. Such symmetry is inherited from the symmetry of the continuous problem and is rooted in the dihedral symmetry \(D_4\) of the domain. Homotopies preserving such symmetry are designed to efficiently compute all solutions of the polynomial systems obtained from the discretizations for problems with cubic and quintic nonlinearities, respectively. The key points in homotopy construction are the special properties of the polynomial systems arising respectively from the discretizations of \(-\Delta u=u^3\) and \(-\Delta u=u^5\) in certain eigensubspaces. Such resulting polynomial systems are taken as start systems in the homotopies. Since only representative solution paths need to be followed, a lot of computational cost can be saved. Numerical results are presented to illustrate the efficiency.