Mesh theorems on traces, normalizations of function traces and their inversion
Mesh theorems on traces, normalizations of function traces and their inversion
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DOI：
10.1515/rnam.1991.6.3.223
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发表时间：
1991
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影响因子：
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通讯作者：
S. V. Nepomnyaschikh
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文献类型：
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作者：
S. V. Nepomnyaschikh
本文研究了二维和三维分片光滑域三角剖分上网格函数的分片线性剖分的迹。特别是，它考虑了压缩三角测量。凝聚三角剖分的方式可以是任意的，但必须满足广泛使用的三角剖分（所谓的正则三角剖分）的三角形假设。本文还分析了网格函数迹空间的等价规范化和这些规范化的逆问题。Sobolev空间的函数迹定理[4]在数学物理边值问题的研究中起着重要的作用。这些定理通常用于推导关于边界条件的稳定性的先验估计。对于网格函数的情况，这个问题的第一个构造性分析似乎是由V. B进行的。Andreev在[1，2]中，其中考虑了矩形网格的情况。在文献[5，6，11，14，18]中考虑了具有混沌节点分布的三角剖分。迹空间中等价规范化的构造与网格函数从一个子域到另一个子域的保范规范化问题密切相关[3，8 - 10，23]。在区域分解方法的发展中，在网格函数的迹空间中构造容易可逆的归一化的问题是特别重要的。这些问题也被认为是在文件中。1.迹的网格定理设是空间R中的区域，η = 2，3，其边界为Γ.我们在定义域Ω上做以下假设。设ε为正参数。然后通过变量的变换，将定义域Ω从满足Lipschitz条件的C类变换为具有分段光滑边界/”的定义域Ω“，且”的性质与ε无关。在η = 3的情况下，我们还将假设在区域Ω的边界上没有圆锥点。设Ω = Ijfi是域Ω的三角剖分[7]。三角测量Ω的节点将表示为7j。我们表示外接在单形t·周围的球的半径，而π表示内接球的半径。我们将假设存在与Ω无关的常数σ，使得r.Jp^a（1.1）和/j，/ = 1，.，.，M是足够小的数，即Ω是正则三角剖分[7]。设F = 3Ω为三角剖分Ω的边界。我们还将假设存在节点z的变换T。三角测量Ω 224 S. V. Nepomnyaschikh使得zie Ω，其中Ω是域Ω的闭包，并且满足以下条件：（a）对于具有顶点z1，Ω的三角剖分Ω的每个单形t*，，zi，其中三角剖分Ω = Uff x i/的单形τ i对应于顶点zi，zi，zi，对于其满足条件（1.1）（可能具有不同的常数σ但与Λ无关）;（B）如果zi ∈ i，则zi，.（c）存在与Ω无关的正常数q和c1，使得对于任何Ω，c1\zi-zj\<\zrzj\<c2\zi-zj\，- 是的€ Ω。这里，是节点z1和z2之间的距离。注1.1.注意，对于要满足的条件（a）-（c），F以二阶精度逼近Γ就足够了。用标准范数表示索伯列夫空间用W^h（&）表示实值连续函数u的空间，这些函数在三角剖分Ω的三角形τ上是线性的，用W^（r）表示从边界F开始的函数的迹的空间：φ”= u u» e到每个节点z· € F，让我们把数fy = \z对应起来。-z/|，其中，是与z1相邻的节点，并且设置
The paper studies traces of piecewise-linear prolongations of mesh functions given on triangulations of twoand three-dimensional piecewise-smooth domains. In particular, it considers condensing triangulations. The way of condensing triangulations may be arbitrary but the widely used assumptions on the triangles of triangulations (the so-called regular triangulations) must be satisfied. Equivalent normalizations of the space of traces of mesh functions and problems of inverting these normalizations are also analysed. The theorems on traces of functions from Sobolev spaces [4] play an important role in studying boundary value problems of mathematical physics. These theorems are commonly used for deriving a priori estimates of the stability with respect to boundary conditions. For the case of mesh functions the first constructive analysis of this problem seems to be carried out by V. B. Andreev in [1,2], where the case of rectangular meshes was considered. Triangulations with chaotic distribution of nodes were considered in [5,6,11,14,18]. The construction of equivalent normalizations in the space of traces is closely related to the problem of norm-preserving prolongations of mesh functions from a subdomain to another subdomain [3,8-10,23]. In connection with the development of the domain decomposition method the problem of construction of easily invertible normalizations in the space of traces of mesh functions is of particular importance. These problems are also considered in the paper. 1. MESH THEOREM ON TRACES Let be a domain in the space R, η = 2,3, with the boundary Γ. We make the following assumptions on the domain Ω. Let ε be a positive parameter. Then making the change of variables the domain Ω is transformed into the domain Ω' with the piecewise-smooth boundary /" from the class C satisfying the Lipschitz condition and the properties of ' are independent of ε. In the case of η = 3 we will also assume that there is no conic points at the boundary of the domain Ω. Let Ω = Ijfi ̂ be a triangulation [7] of the domain Ω. The nodes of the triangulation Ω will be denoted by z^ ί = Ι,.,.,ΛΛ By 7j. we denote the radius of the ball circumscribed around the simplex t·., while by pi we denote the radius of the inscribed ball. We will assume that there exists a constant σ independent of Ω such that r.Jp^a (1.1) and /j, / = Ι,.,.,Μ, are sufficiently small numbers, i.e. Ω is a regular triangulation [7]. Let F = 3Ω be the boundary of the triangulation Ω. We will also assume that there exists a transformation Τ of the nodes z. of the triangulation Ω 224 S. V. Nepomnyaschikh such that zi e Ω, where Ω is the closure of the domain Ω, and that the following conditions are satisfied: (a) to each simplex t*, of the triangulation Ω with the vertices z^ , ζ. , zi there corresponds the simplex τί of the triangulation Ω = Uff x i/ with the vertices zi , zi , zi , for which condition (1.1) is satisfied (possibly with a different constant σ but independent of Λ); (b) if zi € Γ, then z,. e Γ; (c) there exist positive constants q and c^ independent of Ω such that cl\zi-zj\<\zrzj\<c2\zi-zj\ for any ζ^.,ζ. € Ω. Here, ^-ζλ is the distance between the nodes z/ and z.. Remark 1.1. Note that for conditions (a)-(c) to be satisfied, it is sufficient that F approximate Γ with the second-order accuracy. Denote by with the standard norm the Sobolev space Denote by W^h(&) the space of real-valued continuous functions u which are linear on the triangles τ·, of the triangulation Ω and by W^(r) the space of traces of functions from at the boundary F: φ" = u u» e To each node z· € F let us put into correspondence the number fy = \z. -z/|, where z € jT is a node neighbouring upon z^, and set