The design and analysis of the Generalized Finite Element Method
The design and analysis of the Generalized Finite Element Method
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DOI：
10.1016/s0045-7825(99)00072-9
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发表时间：
2000-01
影响因子：
7.2
通讯作者：
T. Strouboulis;I. Babuska;K. Copps
中科院分区：
文献类型：
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作者：
T. Strouboulis;I. Babuska;K. Copps
本文介绍了广义有限元方法，它是经典有限元方法和单位分割法的结合。通过增加反映边值问题和输入数据的已知信息的特殊函数来扩充标准有限元空间，例如，由精确解在角点附近的局部渐近展开得到的奇异函数等，这些特殊函数与对应于标准线性顶点形函数的单位分解相乘，并与已有的有限元基粘贴在一起，构成增广的协调有限元空间。这样，特殊函数所提供的局部可逼近性被包含在近似中，同时保持了现有的有限元代码的基础结构。广义有限元的主要特点是：(1)基本边界条件可以像标准有限元方法一样施加，不像其他基于单位分解的方法，这是一个主要问题；(2)刚度矩阵和载荷向量的数值积分的精度得到自适应控制，以便特殊函数的积分误差不会影响构造的近似精度(这个问题在其他单位分解或无网格法的实现中也没有得到充分解决)；以及(3)通过简单地修改直接线性求解器来解决方程组中的线性相关性。数值算例表明，与标准有限元相比，广义有限元在求解复杂几何区域问题时具有更小的误差和更少的计算机资源。
In this paper, we introduce the Generalized Finite Element Method (GFEM) as a combination of the classical Finite Element Method (FEM) and the Partition of Unity Method (PUM). The standard finite element spaces are augmented by adding special functions which reflect the known information about the boundary value problem and the input data; e.g., the singular functions obtained from the local asymptotic expansion of the exact solution in the neighborhood of a corner point, etc. The special functions are multiplied with the partition of unity corresponding to the standard linear vertex shape-functions and pasted together with the existing finite element basis to construct an augmented conforming finite element space. In this way, the local approximability afforded by the special functions is included in the approximation, while maintaining the existing infrastructure of finite element codes. The major features of the GFEM are: (1) the essential boundary conditions can be imposed exactly as in the standard FEM, unlike other partition of unity based methods where this is a major issue; (2) the accuracy of the numerical integration of the entries of the stiffness matrix and load vector is controlled adaptively so that the errors in integration of the special functions do not affect the accuracy of the constructed approximation (this issue also has not been sufficiently addressed in other implementations of partition of unity or meshless methods); and (3) linear dependencies in the system of equations are resolved by employing an easy modification of the direct linear solver. The power of the GFEM for solving problems in domains with complex geometry with less error and less computer resources than the standard FEM is illustrated by numerical examples.