### 数学代写|计算方法代写computational method代考|Approximate solutions

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## 数学代写|计算方法代写computational method代考|Approximate solutions

The trial and test spaces defined in the preceding section are infinite-dimensional, that is, they span infinitely many linearly independent functions. To find an approximate solution, we construct finite-dimensional subspaces denoted, respectively, by $S \subset X, V \subset Y$ and seek the function $u \in S$ that satisfies $B(u, v)=F(v)$ for all $v \in V$. Let us return to the introductory example described in Section $1.1$ and define
$$u=u_{n}=\sum_{j=1}^{n} a_{j} \varphi_{j}, \quad v=v_{n}=\sum_{i=1}^{n} b_{i} \varphi_{i}$$
where $\varphi_{i}(i=1,2, \ldots n)$ are basis functions. Using the definitions of $k_{i j}$ and $m_{i j}$ given in eq. (1.12), we write the bilinear form as
\begin{aligned} B(u, v) \equiv \int_{0}^{t}\left(\kappa u^{\prime} v^{\prime}+c u v\right) d x &=\sum_{i=1}^{n} \sum_{j=1}^{n}\left(k_{i j}+m_{i j}\right) a_{j} b_{i} \ &={b}^{T}([K]+[M]){a} \end{aligned}
Similarly,
$$F(v) \equiv \int_{0}^{\ell} f v d x=\sum_{i=1}^{n} b_{i} r_{i}={b}^{T}{r}$$
where $r_{i}$ is defined in eq. (1.12). Therefore we can write $B(u, v)-F(v)=0$ in the following form:
$${b}^{T}(([K]+[M]){a}-{r})=0 .$$
Since this must hold for any choice of ${b}$, it follows that
$$([K]+[M]){a}={r}$$

which is the same system of linear equations we needed to solve when minimizing the integral I, see eq. (1.14). Of course, this is not a coincidence. The solution of the generalized problem: “Find $u_{n} \in S$ such that $B\left(u_{n}, v\right)=F(v)$ for all $v \in V^{n}$, minimizes the error in the energy norm. See Theorem 1.4.

Theorem $1.3$ The error $e$ defined by $e=u-u_{n}$ satisfies $B(e, v)=0$ for all $v \in S^{0}(I)$. This result follows directly from
$$\begin{array}{rlr} B(u, v)=F(v) & \text { for all } v \in S^{0}(I) \ B\left(u_{n}, v\right)=F(v) & \text { for all } v \in S^{0}(I) . \end{array}$$
Subtracting the second equation from the first we have,
$$B\left(u-u_{n}, v\right) \equiv B(e, v)=0 \quad \text { for all } v \in S^{0}(I) .$$
This equation is known as the Galerkin ${ }^{11}$ orthogonality condition.
Theorem 1.4 If $u_{n} \in S^{0}(I)$ satisfies $B\left(u_{n}, v\right)=F(v)$ for all $v \in S^{0}(I)$ then $u_{n}$ minimizes the error $u_{E X}-u_{n}$ in energy norm where $u_{E X}$ is the exact solution:
$$\left|u_{E X}-u_{n}\right|_{E(I)}=\min {u \in \bar{S}}\left|u{E X}-u\right|_{E(I)} \text {. }$$
Proof: Let $e=u-u_{n}$ and let $v$ be an arbitrary function in $S^{0}(I)$. Then
$$|e+v|_{E(l)}^{2} \equiv \frac{1}{2} B(e+v, e+v)=\frac{1}{2} B(e, e)+B(e, v)+\frac{1}{2} B(v, v) .$$
The first term on the right is $|e|_{E(I)}^{2}$, the second term is zero on account of Theorem $1.3$, the third term is positive for any $v \neq 0$ in $S^{\circ}(I)$. Therefore $|e|_{E(I)}$ is minimum.

Theorem $1.4$ states that the error depends on the exact solution of the problem $u_{E X}$ and the definition of the trial space $\bar{S}(I)$.

The finite element method is a flexible and powerful method for constructing trial spaces. The basic algorithmic structure of the finite element method is outlined in the following sections.

## 数学代写|计算方法代写computational method代考|The standard polynomial space

The standard polynomial space of degree $p$, denoted by $S^{p}\left(I_{\text {st }}\right)$, is spanned by the monomials $1, \xi, \xi^{2}, \ldots, \xi^{p}$ defined on the standard element
$$I_{\text {st }}={\xi \mid-1<\xi<1} .$$
The choice of basis functions is guided by considerations of implementation, keeping the condition number of the coefficient matrices small, and personal preferences. For the symmetric positive-definite matrices considered here the condition number $C$ is the largest eigenvalue divided by the smallest. The number of digits lost in solving a linear problem is roughly equal to $\log _{10} C$. Characterizing the condition number as being large or small should be understood in this context. In the finite element method the condition number depends on the choice of the basis functions and the mesh.

The standard polynomial basis functions, called shape functions, can be defined in various ways. We will consider shape functions based on Lagrange polynomials and Legendre ${ }^{12}$ polynomials. We will use the same notation for both types of shape function.

Lagrange shape functions
Lagrange shape functions of degree $p$ are constructed by partitioning $I_{\mathrm{st}}$ into $p$ sub-intervals. The length of the sub-intervals is typically $2 / p$ but the lengths may vary. The node points are $\xi_{1}=-1$, $\xi_{2}=1$ and $-1<\xi_{3}<\xi_{4}<\cdots<\xi_{p+1}<1$. The $i$ th shape function is unity in the $i$ th node point and is zero in the other node points:
$$N_{i}(\xi)=\prod_{\substack{k=1 \ k \neq i}}^{p+1} \frac{\xi-\xi_{k}}{\xi_{i}-\xi_{k}}, \quad i=1,2, \ldots, p+1, \quad \xi \in I_{\mathrm{st}}$$
These shape functions have the following important properties:
$$N_{i}\left(\xi_{j}\right)=\left{\begin{array}{ll} 1 & \text { if } i=j \ 0 & \text { if } i \neq j \end{array} \quad \text { and } \sum_{i=1}^{p+1} N_{i}(\xi)=1 .\right.$$
For example, for $p=2$ the equally spaced node points are $\xi_{1}=-1, \xi_{2}=1, \xi_{3}=0$. The corresponding Lagrange shape functions are illustrated in Fig. 1.3.

## 数学代写|计算方法代写computational method代考|Finite element spaces in one dimension

We are now in a position to provide a precise definition of finite element spaces in one dimension.
The domain $I={x \mid 0<x<\ell}$ is partitioned into $M$ non-overlapping intervals called finite elements. A partition, called finite element mesh, is denoted by $\Delta$. Thus $M=M(\Delta)$. The boundary points of the elements are the node points. The coordinates of the node points, sorted in ascending order, are denoted by $x_{i},(i=1,2, \ldots, M+1)$ where $x_{1}=0$ and $x_{M+1}=\ell$. The $k$ th element $I_{k}$ has the boundary points $x_{k}$ and $x_{k+1}$, that is, $I_{k}=\left{x \mid x_{k}<x<x_{k+1}\right}$.

Various approaches are used for the construction of sequences of finite element mesh. We will consider four types of mesh design:

1. A mesh is uniform if all elements have the same size. On the interval $I=(0, \ell)$ the node points are located as follows:
$$x_{k}=(k-1) \ell / M(\Delta) \text { for } k=1,2,3, \ldots, M(\Delta)+1 .$$
2. A sequence of meshes $\Delta_{K}(K=1,2, \ldots)$ is quasiuniform if there exist positive constants $C_{1}, C_{2}$, independent of $K$, such that
$$C_{1} \leq \frac{\ell_{\max }^{(K)}}{\ell_{\min }^{(K)}} \leq C_{2}, \quad K=1,2, \ldots$$
where $\ell_{\max }^{(K)}$ (resp. $\ell_{\min }^{(K)}$ ) is the length of the largest (resp. smallest) element in mesh $\Delta_{K}$. In two and three dimensions $\ell_{k}$ is defined as the diameter of the $k$ th element, meaning the diameter of the smallest circle or sphere that envelopes the element. For example, a sequence of quasiuniform meshes would be generated in one dimension if, starting from an arbitrary mesh, the elements would be successively halved.
3. A mesh is geometrically graded toward the point $x=0$ on the interval $0<x<\ell$ if the node points are located as follows:
$$x_{k}= \begin{cases}0 & \text { for } k=1 \ q^{M(\Delta)+1-k} \ell & \text { for } k=2,3, \ldots, M(\Delta)+1\end{cases}$$
where $0<q<1$ is called grading factor or common factor. These are called geometric meshes.

## 数学代写|计算方法代写computational method代考|Finite element spaces in one dimension

1. 如果所有元素的大小相同，则网格是均匀的。在区间一世=(0,ℓ)节点位置如下：
Xķ=(ķ−1)ℓ/米(Δ) 为了 ķ=1,2,3,…,米(Δ)+1.
2. 一系列网格Δķ(ķ=1,2,…)如果存在正常数是准均匀的C1,C2， 独立于ķ, 这样
C1≤ℓ最大限度(ķ)ℓ分钟(ķ)≤C2,ķ=1,2,…
在哪里ℓ最大限度(ķ)（分别。ℓ分钟(ķ)) 是网格中最大（或最小）元素的长度Δķ. 在二维和三维ℓķ被定义为直径ķth 元素，表示包围该元素的最小圆或球体的直径。例如，如果从任意网格开始，元素将连续减半，则将在一维中生成一系列准均匀网格。
3. 网格朝向该点进行几何分级X=0在区间0<X<ℓ如果节点点的位置如下：
Xķ={0 为了 ķ=1 q米(Δ)+1−ķℓ 为了 ķ=2,3,…,米(Δ)+1
在哪里0<q<1称为分级因子或公因子。这些被称为几何网格。
4. 一个网格是一个激进的网格，如果在区间上01,ķ=1,2,…,米(Δ)+1.可以根据有关确切解决方案的规律性
和实施方面的先验信息来回答在特定应用中优先选择这些方案中的哪一个的问题。指导有限元网格选择的实际考虑将在 1.5.2 节中讨论。

## 有限元方法代写

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## MATLAB代写

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