# 数学代写|有限元方法代写Finite Element Method代考|MECH ENG 4118

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## 数学代写|有限元方法代写Finite Element Method代考|Space-Time Coupled Classical Methods of Approximation

In this chapter we consider space-time coupled classical methods of approximation for initial value problems (IVPs) in which the space-time domain $\bar{\Omega}{x t}$ of the IVP is not discretized. By space-time coupled methods we mean concurrent dependence of all quantities of interest on spatial coordinates as well as time, which is in agreement with the physics of the evolution described by the governing differential equations (GDEs) constituting the IVPs. In order to present development of a general mathematical framework for space-time classical methods of approximation for all IVPs regardless of their origin or field of application, we must consider mathematical classification of all space-time differential operators into distinct categories (Chapter 2), non-self-adjoint and non-linear, and then undertake development of the methods of approximation for these using nondiscretized space-time domain $\bar{\Omega}{x t}$.

We consider space-time classical methods of approximation for the IVP $A \phi-f=0$ in $\Omega_{x t}=\Omega_{x} \times \Omega_{t}$, the entire space-time domain. In spacetime classical methods, the space-time domain $\bar{\Omega}{x t}$ is not discretized. In particular in this chapter we consider space-time methods of approximation that are based on the space-time integral form associated with the IVP $A \phi-f=0$ in $\Omega{x t}$. Since these methods form the foundation of spacetime finite element method, these methods are of special importance and interest. The integral form associated with the IVP $A \phi-f=0$ in $\Omega_{x t}=$ $\Omega_{x} \times \Omega_{t}$ can be constructed either by using the fundamental lemma of the calculus of variations (Chapter 2) or directly by constructing a functional such as residual functional and then setting its first variation to zero. The methods of approximation can be considered using both approaches. The first approach gives rise to space-time Galerkin method (STGM), space-time Galerkin method with weak form (STGM/WF), space-time Petrov-Galerkin method (STPGM), space-time weighted residual method (STWRM), and so on, whereas the second approach is considered in the space-time least squares method or process (STLSM or STLSP). In this chapter we consider all of these space-time methods of approximation.

## 数学代写|有限元方法代写Finite Element Method代考|Space-time integral forms based on fundamental lemma

The space-time integral form associated with the IVP can be established using the fundamental lemma of the calculus of variations (see Chapter 2). If $A \phi-f=0$ in $\Omega_{x t}=\Omega_{x} \times \Omega_{t}$ is the IVP, then based on the fundamental lemma of the calculus of variations
$$\int_{\bar{\Omega}{x t}}(A \phi-f) v d \Omega{x t}=0$$
holds provided $v=0$ on $\Gamma^{}$ if $\phi=\phi_{0}$ on $\Gamma^{}$. Furthermore, $v=\delta \phi$ is admissible due to the fact that it satisfies the condition $v=0$ on $\Gamma^{}$ when $\phi=\phi_{0}$ on $\Gamma^{}$.

Based on (3.1), various space-time classical methods of approximation can be considered. We remark that in considering the classical methods of approximation, it is prudent to consider the space-time domain $\bar{\Omega}{x t}$ instead of $\bar{\Omega}{x t}^{(n)}$, the space-time domain of $n^{t h}$ space-time strip or slab. This is due to the fact that the initial conditions for $\bar{\Omega}{x t}$ are generally simpler whereas ICs for $\bar{\Omega}{x t}^{(n)}$ are extracted from the immediately preceding space-time strip or slab. These ICs must be satisfied by the approximation and hence the choice of $\bar{\Omega}{x t}$ is meritorious over $\bar{\Omega}{x t}^{(n)}$. In what follows we consider the IVP $A \phi-f=0$ in $\bar{\Omega}_{x t}$ to discuss various space-time methods of approximation: STGM, STPGM, STWRM, STGM/WF.

# 有限元方法代考

## 数学代写|有限元方法代写Finite Element Method代考|Space-time integral forms based on fundamental lemma

$$\int_{\bar{\Omega} x t}(A \phi-f) v d \Omega x t=0$$

## 有限元方法代写

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