Algorithms and Visualization for solutions of nonlinear Elliptic equations
Algorithms and Visualization for solutions of nonlinear Elliptic equations
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DOI：
10.1142/s0218127400001006
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发表时间：
2000-07
期刊：
影响因子：
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通讯作者：
Goong Chen;Jianxin Zhou;W. Ni
中科院分区：
文献类型：
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作者：
Goong Chen;Jianxin Zhou;W. Ni
本文利用齐次Dirichlet边界条件计算并可视化了几类主要的半线性椭圆型边值问题的解。提出了山路算法(MPA)、尺度迭代算法(SIA)、单调迭代算法和直接迭代算法(MIA和DIA)。众所周知，半线性椭圆型方程的多解性是众所周知的。许多这样的物理意义重大的解决方案也是众所周知的缺乏稳定性，因此很难用数字来捕捉。我们将计算并可视化这种多重解的轮廓，从而展示结构域对多解性的几何效应。特别强调了SIA和MPA，并用它们计算了多个不稳定解。这些域包括圆盘、对称或非对称环、哑铃和带空洞的哑铃。非线性偏微分方程组包括Lane-Emden方程、Chandrasekhar方程、Henon方程、奇摄动方程和次线性增长方程。相关的数值解数据被列为其他研究人员可能参考的基准。在适当的情况下，将对现有文献中关于解决行为的评论进行说明。文中还将给出由可视化得到的解的一些进一步的理论性质。
In this paper, we compute and visualize solutions of several major types of semilinear elliptic boundary value problems with a homogeneous Dirichlet boundary condition in 2D. We present the mountain–pass algorithm (MPA), the scaling iterative algorithm (SIA), the monotone iteration and the direct iteration algorithms (MIA and DIA). Semilinear elliptic equations are well known to be rich in their multiplicity of solutions. Many such physically significant solutions are also known to lack stability and, thus, are elusive to capture numerically. We will compute and visualize the profiles of such multiple solutions, thereby exhibiting the geometrical effects of the domains on the multiplicity. Special emphasis is placed on SIA and MPA, by which multiple unstable solutions are computed. The domains include the disk, symmetric or nonsymmetric annuli, dumbbells, and dumbbells with cavities. The nonlinear partial differential equations include the Lane–Emden equation, Chandrasekhar's equation, Henon's equation, a singularly perturbed equation, and equations with sublinear growth. Relevant numerical data of solutions are listed as possible benchmarks for other researchers. Commentaries from the existing literature concerning solution behavior will be made, wherever appropriate. Some further theoretical properties of the solutions obtained from visualization will also be presented.