THE P-VERSION OF THE FINITE-ELEMENT METHOD
THE P-VERSION OF THE FINITE-ELEMENT METHOD
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DOI：
10.1137/0718033
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发表时间：
1981-01-01
影响因子：
2.9
通讯作者：
KATZ, IN
中科院分区：
文献类型：
--
作者：
BABUSKA, I;SZABO, BA;KATZ, IN
在有限元方法的p版本中，三角剖分是固定的，分段多项式近似的度p逐渐增加，直到达到所需的精度水平。在本文中，我们首先建立了有限元三角剖分上定义的分段多项式的一些空间的基本近似性质。这些属性导致对 p 版本的渐近收敛率的先验估计。估计表明，当采用准均匀三角剖分并且比较的基础是自由度时，p版本给出的结果并不比传统有限元方法（称为h版本，其中h表示单元的最大直径）所获得的结果差。此外，在奇点问题的情况下，我们表明（在实践中通常满足的条件下）p 版本的收敛速度是具有准均匀网格的 h 版本的两倍。逆近似定理根据固定三角剖分上的分段多项式逼近函数的速率来确定函数的平滑度，并针对奇异和非奇异问题进行了证明。我们提出了数值示例，说明了 p 版本对于简单的一维问题和二维弹性中的两个问题的有效性。我们还讨论了与 p 版本相关的舍入误差和计算成本。最后，我们描述了一些重要的特征，例如分层基函数，这些特征已在 COMET-X（p 版本的实验计算机实现）中使用。
In thep-version of the finite element method, the triangulation is fixed and the degreep, of the piecewise polynomial approximation, is progressively increased until some desired level of precision is reached.In this paper, we first establish the basic approximation properties of some spaces of piecewise polynomials defined on a finite element triangulation. These properties lead to an a priori estimate of the asymptotic rate of convergence of thep-version. The estimate shows that thep-version gives results which are not worse than those obtained by the conventional finite element method (called theh-version, in whichhrepresents the maximum diameter of the elements), when quasi-uniform triangulations are employed and the basis for comparison is the number of degrees of freedom. Furthermore, in the case of a singularity problem, we show (under conditions which are usually satisfied in practice) that the rate of convergence of thep-version is twice that of theh-version with quasi-uniform mesh. Inverse approximation theorems which determine the smoothness of a function based on the rate at which it is approximated by piecewise polynomials over a fixed triangulation are proved for both singular and nonsingular problems.We present numerical examples which illustrate the effectiveness of thep-version for a simple one-dimensional problem and for two problems in two-dimensional elasticity. We also discuss roundoff error and computational costs associated with thep-version. Finally, we describe some important features, such as hierarchic basis functions, which have been utilized in COMET-X, an experimental computer implementation of thep-version.