### 数学代写|有限元方法代写Finite Element Method代考|Find 2022

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|有限元方法代写Finite Element Method代考|General overview

The physical processes encountered in all branches of sciences and engineering can be classified into two major categories: time-dependent processes and stationary processes. Time-dependent processes describe evolutions in which quantities of interest change with time. If the quantities of interest cease to change in an evolution then the evolution is said to have reached a stationary state. Not all evolutions have stationary states. The evolutions without a stationary state are often referred to as unsteady processes. Stationary processes are those in which the quantities of interest do not depend upon time. For a stationary process to be valid or viable, it must correspond to the stationary state of an evolution. Every process in nature is an evolution. Nonetheless it is sometimes convenient to consider their stationary state. In this book we only consider non-stationary processes, i.e. evolutions that may have a stationary state or may be unsteady.

A mathematical description of most stationary processes in sciences and engineering often leads to a system of ordinary or partial differential equations. These mathematical descriptions of the stationary processes are referred to as boundary value problems (BVPs). Since stationary processes are independent of time, the partial differential equations describing their behavior only involve dependent variables and space coordinates as independent variables. On the other hand, mathematical descriptions of evolutions lead to partial differential equations in dependent variables, space coordinates, and time and are referred to as initial value problems (IVPs).

In case of simple physical systems, the mathematical descriptions of IVPs may be simple enough to permit analytical solutions. However, most physical systems of interest may be quite complicated and their mathematical description (IVPs) may be complex enough not to permit analytical solutions. In such cases, two alternatives are possible. In the first case, one could undertake simplifications of the mathematical descriptions to a point that analytical solutions are possible. In this approach, the simplified forms may not be descriptive of the actual behavior and sometimes this simplification may not be possible at all. In the second alternative, we abandon the possibility of theoretical solutions altogether as viable means of solving complex practical problems involving IVPs and instead we resort to numerical meth-ods for obtaining numerical solutions of IVPs. The finite element method (FEM) is one such method of solving IVPs numerically and constitutes the subject matter of this book. Before we delve deeper into the FEM for IVPs, it is perhaps fitting to discuss a little about the broader classes of available methods for obtaining numerical solutions of IVPs.

## 数学代写|有限元方法代写Finite Element Method代考|Space-time coupled methods of approximation

We note that since $\phi=\phi(x, t)$, the solution exhibits simultaneous dependence on spatial coordinates $x$ and time $t$. This feature is intrinsic in the mathematical description (1.1) of the physics.

Thus, the most rational approach to undertake for the solution of (1.1) (approximate or otherwise) is to preserve simultaneous dependence of $\phi$ on $x$ and $t$. Such methods are known as space-time coupled methods. Broadly speaking, in such methods time $t$ is treated as another independent variable in addition to spatial coordinates. Fig. $1.1$ shows space-time domain $\bar{\Omega}{x t}=$ $\Omega{x t} \cup \Gamma ; \Gamma=\cup_{i=1}^{4} \Gamma_{i}$ with closed boundary $\Gamma$. For the sake of discussion, as an example we could have a boundary condition $(\mathrm{BC})$ at $x=0 \forall t \in[0, \tau]$, boundary $\Gamma_{1}$, as well as at $x=L \forall t \in[0, \tau]$, boundary $\Gamma_{2}$, and an initial condition $(\mathrm{IC})$ at $t=0 \forall x \in[0, L]$, boundary $\Gamma_{3}$. Boundary $\Gamma_{4}$ at final value of time $t=\tau$ is open, i.e. at this boundary only the evolution (the solution of (1.1) subjected to these $\mathrm{BCs}$ and $\mathrm{IC})$, will yield the function $\phi(x, \tau)$ and its spatial and time derivatives.

When the initial value problem contains two spatial coordinates, we have space-time slab $\bar{\Omega}{x t}$ shown in Fig. $1.2$ in which $$\Omega{x t}=\left(0, L_{1}\right) \times\left(0, L_{2}\right) \times(0, \tau)$$
is a prism. In this case $\Gamma_{1}, \Gamma_{2}, \Gamma_{3}$, and $\Gamma_{4}$ are faces of the prism (surfaces). For illustration, the possible choices of BCs and ICs could be: BCs on $\Gamma_{1}=$ $A D D_{1} A_{1}$ and $\Gamma_{2}=B C C_{1} B_{1}$, IC on $\Gamma_{3}=A B C D$, and $\Gamma_{4}=A_{1} B_{1} C_{1} D_{1}$ is the open boundary. This concept of space-time slab can be extended for three spatial dimensions and time. Using space-time domain shown in Fig. $1.1$ or $1.2$ and treating time as another independent variable, we could consider the following methods of approximation.

## 数学代写|有限元方法代写Finite Element Method代考|Space-time coupled methods of approximation

$x=0 \forall t \in[0, \tau]$ ，边界 $\Gamma_{1}$ ，以及在 $x=L \forall t \in[0, \tau]$, 边界 $\Gamma_{2}$ ，和一个初始条件 (IC)在 $t=0 \forall x \in[0, L]$ ，边 界 $\Gamma_{3}$. 边界 $\Gamma_{4}$ 在最终时间值 $t=\tau$ 是开放的，即在这个边界上只有演化 ( (1.1) 的解受到这些BCs和IC)，将产 生函数 $\phi(x, \tau)$ 及其空间和时间导数。

$$\Omega x t=\left(0, L_{1}\right) \times\left(0, L_{2}\right) \times(0, \tau)$$

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