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# 数学代写|有限元方法作业代写finite differences method代考|Further topics in the finite element method

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## 数学代写|有限元方法作业代写finite differences method代考|The variational approach

We seek a finite element solution of the problem given by eqns (3.30)-(3.32), viz.
$-\operatorname{div}(k \operatorname{grad} u)=f(x, y) \quad$ in $D$,
with the Dirichlet boundary condition
$$u=g(s) \text { on } C_{1}$$
and the Robin boundary condition
$$k(s) \frac{\partial u}{\partial n}+\sigma(s) u=h(s) \quad \text { on } C_{2}$$
The functional for this problem is found from eqn (2.44) as
$$I[u]=\iint_{D}\left{k\left(\frac{\partial u}{\partial x}\right)^{2}+k\left(\frac{\partial u}{\partial y}\right)^{2}-2 u f\right} d x d y+\int_{C_{2}}\left(\sigma u^{2}-2 u h\right) d s$$
and the solution, $u$, of eqns (5.1)-(5.3) is that function, $u_{0}$, which minimizes $I[u]$ subject to the essential boundary condition $u_{0}=g(s)$ on $C_{1}$.

## 数学代写|有限元方法作业代写finite differences method代考|Collocation and least squares methods

Recall the weighted residual method (Section 2.3) for the solution of
$$\mathcal{L} u=f \quad \text { in } D$$
subject to the boundary condition
$$\mathcal{B} u=b \text { on } C .$$
Define the residual
$$r_{1}(\tilde{u})=\mathcal{L} \tilde{u}-f$$
and the boundary residual
$$r_{2}(\tilde{u})=\mathcal{B} \tilde{u}-b$$
then eqn (2.23) suggests the following general weighted residual equations:
$$\iint_{D} r_{1} v_{i} d x d y+\oint_{C_{2}} r_{2} v_{i} d s=0, \quad i=1, \ldots, n$$
where $\left{v_{i}\right}$ is a set of linearly independent weighting functions which satisfy $v_{i} \equiv 0$ on $C_{1}$, that part of $C$ on which an essential boundary condition applies. The trial functions $\tilde{u}$ are defined in the usual piecewise sense by eqn (5.4) as
$$\tilde{u}=\sum_{e} \tilde{u}^{e},$$
with $\tilde{u}^{e}$ interpolated through element $[e]$ in terms of the nodal values. The equations (5.19) then yield a set of algebraic equations for these nodal values. Notice that no restriction is placed on the operator $\mathcal{L}$; it may be non-linear, in which case the resulting set of equations is a non-linear algebraic set; see Section 5.3. Very often, the equations (5.19) are transformed by the use of an integrationby-parts formula, Green’s theorem, so that the highest-order derivative occurring in the integrand is reduced, thus reducing the continuity requirement for the chosen trial function.

## 数学代写|有限元方法作业代写FINITE DIFFERENCES METHOD代考|Use of Galerkin’s method for time-dependent and non-linear problems

When the finite element method is applied to time-dependent problems, the time variable is usually treated in one of two ways:
(1) Time is considered as an extra dimension, and shape functions in space and time are used. This is illustrated in Example 5.3.
(2) The nodal variables are considered as functions of time, and the space variables are used in the finite element analysis.

## 数学代写|有限元方法作业代写FINITE DIFFERENCES METHOD代考|THE VARIATIONAL APPROACH

−div⁡(ķ毕业⁡在)=F(X,是)在D,

ķ(s)∂在∂n+σ(s)在=H(s) 在 C2

I[u]=\iint_{D}\left{k\left(\frac{\partial u}{\partial x}\right)^{2}+k\left(\frac{\partial u}{\部分 y}\right)^{2}-2 u f\right} d x d y+\int_{C_{2}}\left(\sigma u^{2}-2 u h\right) d sI[u]=\iint_{D}\left{k\left(\frac{\partial u}{\partial x}\right)^{2}+k\left(\frac{\partial u}{\部分 y}\right)^{2}-2 u f\right} d x d y+\int_{C_{2}}\left(\sigma u^{2}-2 u h\right) d s

## 数学代写|有限元方法作业代写FINITE DIFFERENCES METHOD代考|COLLOCATION AND LEAST SQUARES METHODS

r1(在~)=大号在~−F

r2(在~)=乙在~−b

∬Dr1在一世dXd是+∮C2r2在一世ds=0,一世=1,…,n

## 数学代写|有限元方法作业代写FINITE DIFFERENCES METHOD代考|USE OF GALERKIN’S METHOD FOR TIME-DEPENDENT AND NON-LINEAR PROBLEMS

1时间被认为是一个额外的维度，并使用空间和时间的形状函数。示例 5.3 对此进行了说明。
2节点变量被认为是时间的函数，空间变量被用于有限元分析。