Failure of the discrete maximum principle for an elliptic finite element problem
Failure of the discrete maximum principle for an elliptic finite element problem
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DOI：
10.1090/s0025-5718-04-01651-5
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发表时间：
2004-03
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影响因子：
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通讯作者：
Andrei Dr;A. Anescu;T. Dupont;L. Ridgway;Scott
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作者：
Andrei Dr;A. Anescu;T. Dupont;L. Ridgway;Scott
一个长期存在的问题是，对于由椭圆型问题的有限元近似引起的离散方程，是否需要一定的网格限制来满足最大条件。这与知道离散格林函数对于三角形网格是否为正有关，允许足够好的近似h1函数。我们研究了用连续分段线性的伽辽金方法离散的二维泊松问题的这个问题。我们给出的例子表明，在一般情况下，答案是否定的，并且我们进一步扩大了已知答案是正的情况的数量。我们的技术利用了一些关于离散格林函数的新结果，这些结果具有独立的意义。
There has been a long-standing question of whether certain mesh restrictions are required for a maximum condition to hold for the discrete equations arising from a finite element approximation of an elliptic problem. This is related to knowing whether the discrete Green's function is positive for triangular meshes allowing sufficiently good approximation of H 1 functions. We study this question for the Poisson problem in two dimensions discretized via the Galerkin method with continuous piecewise linears. We give examples which show that in general the answer is negative, and furthermore we extend the number of cases where it is known to be positive. Our techniques utilize some new results about discrete Green's functions that are of independent interest.