The finite element method
The finite element method
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DOI：
10.1007/978-1-4612-0575-3_12
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发表时间：
1998
期刊：
影响因子：
--
通讯作者：
B. Reddy
中科院分区：
文献类型：
--
作者：
B. Reddy
在实际情况下，确定用于Galerkin方法的合适的基函数可能非常困难，特别是在域Ω不具有简单形状的情况下。有限元法克服了这个困难，提供了一个系统的手段，产生基函数的域上相当任意的形状。使该方法特别吸引人的是，这些基函数是仅在Ω的相对小的部分上非零的分段多项式，使得计算可以以模块化的方式进行，这非常适合于基于计算机的方法。正如我们在后面将要指出的，由有限元方法定义的空间族Vh（h∈（0，1））在适当的意义下具有当h趋于零时Vh趋于Vh的性质。这当然是Galerkin方法收敛的一个不可缺少的性质。
In practical situations the determination of suitable basis functions for use in the Galerkin method can be extremely difficult, especially in cases for which the domain Ω does not have a simple shape. The finite element method overcomes this difficulty by providing a systematic means for generating basis functions on domains of fairly arbitrary shape. What makes the method especially attractive is the fact that these basis functions are piecewise polynomials that are nonzero only on a relatively small part of Ω, so that computations may be carried out in a modular fashion, which is well suited to computer-based approaches. As we show a little later, the family of spacesVh(h∈ (0, 1)) defined by the finite element procedure possesses the property thatVhapproachesVas h approaches zero, in an appropriate sense. This is, of course, an indispensable property for convergence of the Galerkin method.