数学代写|CE595 Finite Element Method

Show that the function $u(r, \theta)=r^{1 / 2} \sin (\theta / 2)$ satisfies Laplace’s equation expressed in polar coordinates;
$$
\frac{\partial^2 u}{\partial r^2}+\frac{1}{r} \frac{\partial u}{\partial r}+\frac{1}{r^2} \frac{\partial^2 u}{\partial \theta^2}=0 .
$$
Given that $u \in \mathcal{H}^s(\Omega) \Longleftrightarrow\left|D^s u\right|<\infty$, show that $u(r, \theta)$ defined on the cracked domain $\Omega$ where $0 \leq r \leq 1$ and $0 \leq \theta \leq 2 \pi$ is in $\mathcal{H}^1(\Omega)$, but is not in $\mathcal{H}^2(\Omega)$. (Hint: work in polar coordinates, and note that $d x d y=r d r d \theta$.)

Given a source function $f \in C^0(\bar{\Omega})$ defined on some open bounded domain $\Omega \subset \mathbb{R}^2$ with boundary $\partial \Omega$, let $u: \Omega \rightarrow \mathbb{R}$ be the solution of the boundary value problem:
$$
\left.\begin{array}{rlr}
-\nabla^2 u+\gamma u=f & & \text { in } \Omega \
u=0 & \text { on } \partial \Omega
\end{array}\right}
$$
where $\gamma>0$ is a given constant.
Given the “test space” $V:=\left{v \mid v \in \mathcal{H}^1(\Omega), v=0\right.$ on $\left.\partial \Omega_D\right}$, where $\mathcal{H}^1(\Omega)$ is the standard Sobolev space, show that $u$ satisfies the variational formulation: find $u \in V$ such that
$$
\int_{\Omega} \nabla u \cdot \nabla v+\gamma \int_{\Omega} u v=\int_{\Omega} v f \quad \text { for all } v \in V .
$$
Show that a weak solution is unique.
Define the Galerkin approximation $u_h \in V_h \subset V$, using a general set of spatial finite element basis functions $\left{\phi_j\right}_{j=1}^k$. Show that the method leads to a $k \times k$ matrix system
$$
A \mathbf{x}=\mathbf{f}
$$
where $A$ is a symmetric matrix with coefficients that you should explicitly identify.
Show further that $\mathbf{x}^T A \mathbf{x} \geq \gamma \mathbf{x}^T M \mathbf{x}$ where $M$ is the standard “mass” matrix associated with the basis set $\left{\phi_j\right}_{j=1}^k$. Hence or otherwise show that $A$ is positive definite.

Show that if a mapped $\boldsymbol{Q}_1$ element is a parallelogram with vertices $\left(x_0, y_0\right)$, $\left(x_0+h_x, y_1\right),\left(x_1+h_x, y_1+h_y\right)$, and $\left(x_1, y_0+h_y\right)$, then the Jacobian $J$ is a constant matrix.

Compute the $4 \times 4$ element stiffness matrix in the case of a rectangular $\boldsymbol{Q}_1$ element (so that $\left.\left(x_0, y_0\right)=\left(x_1, y_0\right)=(0,0)\right)$, and show that if the aspect ratio is big enough then the Galerkin matrix associated with a uniform grid of stretched rectangles has at least one positive off-diagonal entry.

Given the Gauss points $\xi_s= \pm 1 / \sqrt{3}$ and $\eta_t= \pm 1 / \sqrt{3}$ as illustrated in Figure 1.15 in [ESW], show that if $f$ is bilinear, that is, $f(\xi, \eta)=$ $(a+b \xi)(c+d \eta)$ where $a, b, c$ and $d$ are constants, then
$$
\int_{-1}^1 \int_{-1}^1 f \mathrm{~d} \xi \mathrm{d} \eta=\sum_s \sum_t f\left(\xi_s, \eta_t\right) .
$$
Give an example of a polynomial function which is not integrated exactly by this Gauss rule but is integrated exactly by the $3 \times 3$ Gauss rule shown in Figure 1.15.