Generalized finite element method for second-order elliptic operators with Dirichlet boundary conditions
Generalized finite element method for second-order elliptic operators with Dirichlet boundary conditions
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DOI：
10.1016/j.cam.2007.04.041
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发表时间：
2008-08
影响因子：
2.4
通讯作者：
I. Babuska;V. Nistor;Nicolae Ţarfulea
中科院分区：
文献类型：
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作者：
I. Babuska;V. Nistor;Nicolae Ţarfulea
我们引入了一种在广义有限元法 (GFEM) 框架内近似基本边界条件（狄利克雷型条件）的方法。我们的结果适用于 Ω 中 [公式：参见文本] 形式的一般椭圆边值问题，∂Ω 上 u=0，其中 Ω 是平滑有界域。作为测试试验空间，我们考虑 GFEM 空间序列 {Sμ}μ⩾1，它们是不合格的（即 Sμ⊄H01(Ω)）。我们假设 [公式：见正文] ，对于所有 v∈Sμ，并且存在 uI∈Sμ，使得 [公式：见正文] , 0⩽j⩽m，其中 u∈Hm+1(Ω) 是精确解，m 是期望的近似阶数，hμ 是定义 Sμ 的元素的典型大小。在这些条件下，我们证明了 GFEM 近似序列 uμεSμof u 的准最优收敛率。接下来，我们将分析扩展到Ω中的非齐次边值问题[公式：见正文]，u=g on ∂Ω。最后，我们概述了满足我们假设的 GFEM 空间序列 Sμ⊂S~μ，μ=1,2，… 的构造。
We introduce a method for approximating essential boundary conditions—conditions of Dirichlet type—within the generalized finite element method (GFEM) framework. Our results apply to general elliptic boundary value problems of the form [Formula: see text] in Ω, u=0 on ∂Ω, where Ω is a smooth bounded domain. As test-trial spaces, we consider sequences of GFEM spaces, {Sμ}μ⩾1, which are nonconforming (that is Sμ⊄H01(Ω)). We assume that [Formula: see text] , for all v∈Sμ, and there exists uI∈Sμsuch that [Formula: see text] , 0⩽j⩽m, where u∈Hm+1(Ω) is the exact solution, m is the expected order of approximation, and hμis the typical size of the elements defining Sμ. Under these conditions, we prove quasi-optimal rates of convergence for the GFEM approximating sequence uμ∈Sμof u. Next, we extend our analysis to the inhomogeneous boundary value problem [Formula: see text] in Ω, u=g on ∂Ω. Finally, we outline the construction of a sequence of GFEM spaces Sμ⊂S˜μ, μ=1,2,…, that satisfies our assumptions.