### 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

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].

∫0吨X2(一种−1)dX>0

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

H=最大限度jℓj/ℓ,j=1,2,…米(Δ)

(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

(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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