计算数学 1979, 1(4) 378-385 DOI:     ISSN: 0254-7791 CN: 11-2125/O1

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PubMed

论间断有限元的理论

冯康

中国科学院计算中心

摘要

有限元法的理论早在六十年代前期即已建立,而且对经典的连续元——协调元——的情况来说,这一理论业已发展到相当完整细密的程度,见,例如[2.3]。有限元方法高度的有效性和普遍性是与它在理论上的牢靠性和彻底性密切联系着的,但是,间断——非协调——有限元的理论则还处在不甚令人满意的状态,尽管也有了若干重要的进展,

关键词

ON THE THEORY OF DISCONTINUOUS FINITE ELEMENTS

Feng Kang Computing Center, Chinese Academy of Sciences

Abstract:

The theory of finite elements has been established since the early sixties and hasbeen developed to a certain degree of completeness and sophistication for the classicalcontinuous (conforming) cose. The theory for the discontinuous (nonconforming)case is still in a less satisfactory state, although important progress has been made.The present work deals with the theoretical foundation of the discontinuous finiteelements. In section 1, Poincare inequalities for discontinuous functions are given. Theydiffer from the classical ones by an additional term of the integral of jump valuessquared with a constant which measures the density of distribution of discontinuities.On this basis, in sections 2 and 3, injection theorems--discrete analogs of the classicalones--for the discontinuous finite element functions spaces can be established for thecase of formal Sobolev norm (discontinuity discarded) as well as for the case of normcentaining additional penalty (counting the discontinuity). For the first case, a cer-tain condition of weak discontinuity is imposed and this condition is satisfied practi-cally by all the non-conforming elements now in use. In the second case, the condi-tion of weak discontinuity may be violated, i.e., the discontinuity may be arbitrarilystrong. This suggests, among others. two kinds of policy for using discontinuouselements: the policy of tolerance--this is the usual method--in ease of weak dis-continuity and the policy of suppression--this is the penalty method--in case ofstrong discontinuity. In section 4 a general convergence theorem of the penalty method for solvingelliptic equations of order 2m is given to the effect that it is always convergentwhen the finite element interpolation operator is exact to the degree k≥m and thepenalty parameters p_i satisfy 1

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收稿日期  修回日期  网络版发布日期 1979-04-14 00:00:00.0 
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