数学代写|计算方法代写computational method代考|Estimation of error in energy norm

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数学代写|计算方法代写computational method代考|Estimation of error in energy norm

数学代写|计算方法代写computational method代考|Estimation of error in energy norm

We have seen that the finite element solution minimizes the error in energy norm in the sense of eq. (1.48). It is natural therefore to use the energy norm as a measure of the error of approximation. There are two types of error estimators: (a) A priori estimators that establish the asymptotic rate of convergence of a discretization scheme, given information about the regularity (smoothness) of the exact solution and (b) a posteriori estimators that provide estimates of the error in energy norm for the finite element solution of a particular problem.

There is a very substantial body of work in the mathematical literature on the a priori estimation of the rate of convergence, given a quantitative measure of the regularity of the exact solution and a sequence of discretizations. The underlying theory is outside of the scope of this book; however, understanding the main results is important for practitioners of finite element analysis. For details we refer to $[28,45,70,84]$.

数学代写|计算方法代写computational method代考|Regularity

Let us consider problems the exact solution of which has the functional form
u_{E X}=x^{\alpha} \varphi(x), \quad \alpha>1 / 2, \quad x \in I=(0, \ell)
where $\varphi(x)$ is an analytic or piecewise analytic function, see Definition A.1 in the appendix. Our motivation for considering functions in this form is that this family of functions models the singular behavior of solutions of linear elliptic boundary value problems near vertices in polygonal and polyhedral domains. For $u_{E X}$ to be in the energy space, its first derivative must be square integrable on $I$. Therefore
\int_{0}^{t} x^{2(\alpha-1)} d x>0
from which it follows that $\alpha$ must be greater than $1 / 2$.
In the following we will see that when $\alpha$ is not an integer then the degree of difficulty associated with approximating $u_{E X}$ by the finite element method is related to the size of $(\alpha-1 / 2)>0$. The smaller $(\alpha-1 / 2)$ is, the more difficult it is to approximate $u_{E X}$.

If $\alpha$ is a fractional number then the measure of regularity used in the mathematical literature is the maximum number of square integrable derivatives, with the notion of derivative generalized to fractional numbers. See sections A.2.3 and A.2.4 in the appendix. For our purposes it is sufficient to remember that if $u_{E X}$ has the functional form of eq. (1.89), and $\alpha$ is not an integer, then $u_{E X}$ lies in the Sobolev space $H^{\alpha+1 / 2-\epsilon}(I)$ where $\epsilon>0$ is arbitrarily small. This means that $\alpha$ must be larger than $1 / 2$ for the first derivative of $u_{E X}$ to be square integrable. See, for example, [59].

If $\alpha$ is an integer then $u_{E X}$ is an analytic or piecewise analytic function and the measure of regularity is the size of the derivatives of $u_{E X}$. Analogous definitions apply to two and three dimensions.
Remark $1.9$ The $k$ th derivative of a function $f(x)$ is a local property of $f(x)$ only when $k$ is an integer. This is not the case for non-integer derivatives.

数学代写|计算方法代写computational method代考|A priori estimation of the rate of convergence

Analysts are called upon to choose discretization schemes for particular problems. A sound choice of discretization is based on a priori information on the regularity of the exact solution. If we know that the exact solution lies in Sobolev space $H^{k}(I)$ then it is possible to say how fast the error in energy norm will approach zero as the number of degrees of freedom is increased, given a scheme by which a sequence of discretizations is generated. Index $k$ can be inferred or estimated from the input data $\kappa, c$ and $f$.
We define
h=\max {j} \ell{j} / \ell, \quad j=1,2, \ldots M(\Delta)
where $\ell_{j}$ is the length of the jth element, $\ell$ is the size of the of the solution domain $I=(1, \ell)$. This is generalized to two and three dimensions where $\ell$ is the diameter of the domain and $\ell_{j}$ is the diameter of the jth element. In this context diameter means the diameter of the smallest circlein one and two dimensions, or sphere in three dimensions,that contains the element or domain. In two and three dimensions the solution domain is denoted by $\Omega$.

The a priori estimate of the relative error in energy norm for $u_{E X} \in H^{k}(\Omega)$, quasiuniform meshes and polynomial degree $p$ is
\left(e_{r}\right){E} \stackrel{\text { def }}{=} \frac{\left|u{E X}-u_{F E}\right|_{E(\Omega)}}{\left|u_{E X}\right|_{E(\Omega)}} \leq \begin{cases}C(k) \frac{h^{k-1}}{p^{k-1}}\left|u_{E X}\right|_{H^{k}(\Omega)} & \text { for } k-1 \leq p \ C(k) \frac{h^{p}}{p^{k-1}}\left|u_{E X}\right|_{H p^{+1}(\Omega)} & \text { for } k-1>p\end{cases}
where $E(\Omega)$ is the energy norm, $k$ is typically a fractional number and $C(k)$ is a positive constant that depends on $k$ but not on $h$ or $p$. This inequality gives the upper bound for the asymptotic rate of convergence of the relative error in energy norm as $h \rightarrow 0$ or $p \rightarrow \infty$ [22]. This estimate holds for one, two and three dimensions. For one and two dimensions lower bounds were proven in $[13,24]$ and [46] and it was shown that when singularities are located in vertex points then the rate of convergence of the $p$-version is twice the rate of convergence of the $h$-version when both are expressed in terms of the number of degrees of freedom. It is reasonable to assume that analogous results can be proven for three dimensions; however, no proofs are available at present.
We will find it convenient to write the relative error in energy norm in the following form
\left(e_{r}\right){E} \leq \frac{C}{N^{\beta}} $$ where $N$ is the number of degrees of freedom and $C$ and $\beta$ are positive constants, $\beta$ is called the algebraic rate of convergence. In one dimension $N \propto 1 / h$ for the $h$-version and $N \propto p$ for the $p$-version. Therefore for $k-1{0}\right|^{\lambda}$ and $x_{0} \in \bar{I}$ is a nodal point then $\beta=2(k-1)$ for the $p$-version: The rate of $p$-convergence is twice that of $h$-convergence [22,84].

数学代写|计算方法代写computational method代考|Estimation of error in energy norm


数学代写|计算方法代写computational method代考|Estimation of error in energy norm

我们已经看到,有限元解决方案在 eq 的意义上最小化了能量范数的误差。(1.48)。因此很自然地使用能量范数作为近似误差的度量。有两种类型的误差估计器:(a)先验估计器,它建立离散化方案的渐近收敛速度,给定关于精确解的规律性(平滑度)的信息;(b)后验估计器,提供对离散化方案的估计特定问题的有限元解的能量范数误差。


数学代写|计算方法代写computational method代考|Regularity

在哪里披(X)是解析或分段解析函数,见附录中的定义 A.1。我们考虑这种形式的函数的动机是,这个函数族模拟了多边形和多面体域中顶点附近的线性椭圆边值问题的解的奇异行为。为了在和X要在能量空间中,它的一阶导数必须是平方可积的一世. 所以
下面我们将看到,当一种不是整数,那么与近似相关的难度在和X由有限元法与尺寸有关(一种−1/2)>0. 越小(一种−1/2)是,越难近似在和X.

如果一种是分数,那么数学文献中使用的规律性度量是平方可积导数的最大数,导数的概念推广到分数。见附录 A.2.3 和 A.2.4 部分。为了我们的目的,记住如果在和X具有 eq 的函数形式。(1.89),和一种不是整数,那么在和X位于索博列夫空间H一种+1/2−ε(一世)在哪里ε>0任意小。这意味着一种必须大于1/2对于一阶导数在和X是平方可积的。例如,参见 [59]。

如果一种那么是整数在和X是解析或分段解析函数,规律性的度量是导数的大小在和X. 类似的定义适用于二维和三维。
评论1.9这ķ函数的 th 导数F(X)是本地财产F(X)只有当ķ是一个整数。对于非整数导数,情况并非如此。

数学代写|计算方法代写computational method代考|A priori estimation of the rate of convergence

要求分析师为特定问题选择离散化方案。离散化的合理选择基于关于精确解的规律性的先验信息。如果我们知道精确解在 Sobolev 空间Hķ(一世)那么可以说随着自由度数量的增加,能量范数的误差将多快接近零,给定一个生成离散序列的方案。指数ķ可以从输入数据中推断或估计ķ,C和F.
在哪里ℓj是第 j 个元素的长度,ℓ是解域的大小一世=(1,ℓ). 这被推广到二维和三个维度,其中ℓ是域的直径和ℓj是第 j 个元素的直径。在此上下文中,直径是指包含元素或域的一维和二维最小圆或三维球的直径。在二维和三维中,解域表示为Ω.

能量范数相对误差的先验估计在和X∈Hķ(Ω), 拟均匀网格和多项式次数p是
(和r)和= 定义 |在和X−在F和|和(Ω)|在和X|和(Ω)≤{C(ķ)Hķ−1pķ−1|在和X|Hķ(Ω) 为了 ķ−1≤p C(ķ)Hppķ−1|在和X|Hp+1(Ω) 为了 ķ−1>p
在哪里和(Ω)是能量范数,ķ通常是小数,并且C(ķ)是一个正常数,取决于ķ但不在H或者p. 这个不等式给出了能量范数中相对误差的渐近收敛率的上限为H→0或者p→∞[22]。这一估计适用于一维、二维和三个维度。对于一维和二维的下界已在[13,24]和 [46] 表明,当奇点位于顶点时,则p-version 是收敛速度的两倍H-当两者都以自由度数表示时的版本。假设可以在三个维度上证明类似的结果是合理的;但是,目前没有证据可用。
(和r)和≤Cñb在哪里ñ是自由度的数量和C和b是正常数,b称为代数收敛速度。在一维ñ∝1/H为了H-版本和ñ∝p为了p-版本。因此对于k-1{0}\right|^{\lambda}k-1{0}\right|^{\lambda}和X0∈一世¯那么是一个节点b=2(ķ−1)为了p-version:速率p-收敛性是 的两倍H-收敛[22,84]。

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术语 广义线性模型(GLM)通常是指给定连续和/或分类预测因素的连续响应变量的常规线性回归模型。它包括多元线性回归,以及方差分析和方差分析(仅含固定效应)。



有限元是一种通用的数值方法,用于解决两个或三个空间变量的偏微分方程(即一些边界值问题)。为了解决一个问题,有限元将一个大系统细分为更小、更简单的部分,称为有限元。这是通过在空间维度上的特定空间离散化来实现的,它是通过构建对象的网格来实现的:用于求解的数值域,它有有限数量的点。边界值问题的有限元方法表述最终导致一个代数方程组。该方法在域上对未知函数进行逼近。[1] 然后将模拟这些有限元的简单方程组合成一个更大的方程系统,以模拟整个问题。然后,有限元通过变化微积分使相关的误差函数最小化来逼近一个解决方案。





随机过程,是依赖于参数的一组随机变量的全体,参数通常是时间。 随机变量是随机现象的数量表现,其时间序列是一组按照时间发生先后顺序进行排列的数据点序列。通常一组时间序列的时间间隔为一恒定值(如1秒,5分钟,12小时,7天,1年),因此时间序列可以作为离散时间数据进行分析处理。研究时间序列数据的意义在于现实中,往往需要研究某个事物其随时间发展变化的规律。这就需要通过研究该事物过去发展的历史记录,以得到其自身发展的规律。


多元回归分析渐进(Multiple Regression Analysis Asymptotics)属于计量经济学领域,主要是一种数学上的统计分析方法,可以分析复杂情况下各影响因素的数学关系,在自然科学、社会和经济学等多个领域内应用广泛。


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