ALTERNATING ITERATIONS OF FINITE ELEMENT APPROXIMATIONS APPLIED TO THE NONLINEAR BOUNDARY VALUE PROBLEM Δu=bu^2
ALTERNATING ITERATIONS OF FINITE ELEMENT APPROXIMATIONS APPLIED TO THE NONLINEAR BOUNDARY VALUE PROBLEM Δu=bu^2
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有限元逼近的交替迭代应用于非线性边值问题Δu=bu^2
DOI：
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发表时间：
1985
期刊：
影响因子：
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通讯作者：
Tetsuro Yamamoto
中科院分区：
文献类型：
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作者：
K. Ishihara;Yasuto Fukunaga;Tetsuro Yamamoto
这里x==（xi, x2，…）， x.)， 9是n维欧几里得n空间R '中的有界凸域，其边界R是分段光滑的，a是拉普拉斯算子（a: 02/Ox?）， i=1 b是一个正常数，给定函数g(x)是光滑非负的。例如，在气体动力学和化学反应中就会出现这样的问题。在这些情况下，未知函数u(x)代表化学物质的浓度，所以u(x)必须是非负的。建立了式（1.1）的非负解的唯一性和存在性[1,11]。在之前的论文[8,9]中，我们考虑了（1.1）的有限元近似，并提出了基于分段线性多项式和分段常数函数的非线性代数方程组的单调迭代求解方法。本文的目的是介绍交替迭代。进一步，我们证明这些迭代产生的近似交替地大于或小于离散问题的解。从计算的角度来看，这是一个明显的优势。最后，通过数值算例验证了交替迭代的有效性。有限差分法的相关结果见[5,6,10]。
Here x==(xi, x2,..., x.), 9 is a bounded convex domain in the n-dimensional Euclidean n space R", its boundary r is piecewise smooth, A is the Laplace operator (A : 2 02/Ox?), i=1 b is a positive constant, and a given function g(x) is smooth and nonnegative. Such problems arise, for example, in gas dynamics and chemical reactions. In these cases, the unknown function u(x) represents the chemical concentration, so that u(x) is required to be nonnegative. The uniqueness and existence ,of the-nonnegative solution of (1.1) was established [1, 11]. In previous papers [8, 9], we considered the finite element approximations for (1.1), and presented the monotone iterative methods for solving a system of nonlinear algebraic equations associated with the finite element schemes based on piecewise linear polynomials and piecewise constant functions.' The objective of this paper is to present alternating iterations. Further, we shovv that these iterations generate approximations alternately greater and less than the solution of the discrete problem. From a computational point of view, this is a definite advantage. Finally, some numerical examples are given to demonstrate the effectiveness of the alternating iterations. For the related results by the finite difference method, see [5, 6, 10].