Computer-assisted enclosure methods for elliptic differential equations
Computer-assisted enclosure methods for elliptic differential equations
复制标题
椭圆微分方程的计算机辅助包围法
DOI：
10.1016/s0024-3795(00)00273-1
复制
发表时间：
2001
影响因子：
1.1
通讯作者：
M. Plum
中科院分区：
文献类型：
--
作者：
M. Plum
我们报告了计算二阶非线性椭圆边值问题解的包围体的方法，同时证明了包围集中解的存在性。老式的“单调性方法”非常适合这项任务，但仅适用于有限的一类问题。因此，我们提出了一种新方法，该方法基于问题的合适定点公式，并使用给定问题在数值计算的近似解 ω 处的线性化逆的范数界限作为基本要素。这些范数界限是通过特征值包围获得的。我们还简要描述了 M.T. 提出的替代方法。中尾。
We report on methods for computing enclosures of solutions of second-order nonlinear elliptic boundary value problems, simultaneously proving the existence of a solution in the enclosing set. The old-fashioned `monotonicity methods' are well suited for this task, but only for a restricted class of problems. Therefore, we propose a new approach which is based on a suitable fixed-point formulation of the problem and uses, as an essential ingredient, norm bounds for the inverse of the linearization of the given problem at some approximate solution ω which is computed numerically. These norm bounds are obtained via eigenvalue enclosures. We also give a brief description of an alternative method proposed by M.T. Nakao.