On a Finite Difference Analogue of an Elliptic Boundary Problem which is Neither Diagonally Dominant Nor of Non‐negative Type
On a Finite Difference Analogue of an Elliptic Boundary Problem which is Neither Diagonally Dominant Nor of Non‐negative Type
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非对角占优非非负型椭圆边界问题的有限差分模拟
DOI：
10.1002/sapm1964431117
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发表时间：
1964
期刊：
影响因子：
--
通讯作者：
B. Hubbard
中科院分区：
文献类型：
--
作者：
J. Bramble;B. Hubbard
1.导论.在椭圆型方程逼近边值问题的有限差分法离散误差的一般研究中，最大值原理起着中心作用。1930年S. Gerschgorin（14）给出了椭圆方程Dirichlet问题解的一类有限差分类似解的收敛阶的估计方法。所得联立方程组的矩阵具有对角优势性质，即每行非对角元的绝对值之和不超过对角元的大小。此外，它还满足了正交元素都是正的，而非正交元素是非正的条件（2.10）。在此之后，其他人（3），（8），（9），（12），（20），（21）在这些条件的框架内推广了Gerschgorm的结果。最近，作者（5）、（6）给出了椭圆型方程III的Dmchlet问题的有限差分类似公式的一个定理，其中在边界附近放松了性质（2.10）。作为定理的例子，讨论了某些高阶有限差分类似物，并证明了它们具有出乎意料的收敛性质。然而，关于入口问题的最大原理的问题已经被提出，并且直到最近才由M Rockoff（18）处理过。本文的目的之一是给出一个具体的例子，其中条件（2.10）在每个内点都不成立，但极大值原理仍然成立。这是在第2节中完成的，其中研究了两点边值问题（2.1）的O（h '）逼近（h是网格常数）。有趣的事实是，近似的算子附近的边界只需要O（h ~ 2），而不破坏问题的整体精度被证明是真实的。
1. Introduction. In the usual study of the discretIzation error resulting from a, pproximatmg boundary problems for ellIptic equations by finite difference methods the maximum prmCIple plays a central role. In 1930 S. Gerschgorin (14) gave a method for estimatmg the order of convergence of the solutIOn to a certam class of fimte difference analogues to the solutIOn of the DirIchlet problem for ellIptic equatIOns. The matrIX of the resultmg system of SImultaneous lmear equatIOns possesses the property of diagonal dominance, ie the sum of the absolute values of the off-dIagonal elements in each row does not exceed the magmtude of the dIagonal element. Furthermore It satIsfies the condItIon that the dIagonal elements are all posItive and the off-dIagonal elements are non-POSItIve (2.10). Following this, others (3),(8),(9),(12),(20),(21) have extended the results of Gerschgorm within the framework of these condItIOns. Recently the authors (5),(6) gave a theorem on the formulatIon of fimte dIfference analogues of the Dmchlet problem for elliptIC equatIOns III whICh the propertIes (2.10) were relaxed near the boundary. As examples of the theorem certain hIgher order fimte dIfference analogues were dIscussed and shown to have convergence properties prevIously unexpected. The question of a maXImum prmciple for the entIre problem was CIrcumvented however and has been dealt WIth only recently by M Rockoff (18). Hence It IS clear that the suffiCIent condI-tIons (2 10) are not necessaryOne purpose of this paper is to give a specific example in which conditions (2.10) are violated at every interIOr point but for which a maximum principle still holds. This is done in section 2 where an O (h') approximation to the two point boundary problem (2.1) is studIed (h is the mesh constant). The interesting fact that the approximation to the operator near the boundary need be only O (h2) without destroymg the overall accuracy of the problem is shown to be true.