Exercise 1.4: Quadrature Error ${\sim 1$ page $}$
Consider the problem
\begin{aligned} -\Delta u & =f \text { in } \Omega=[0,1] \times[0,1] \ u & =g_D \text { on } \partial \Omega, \end{aligned}
for a prescribed boundary datum $g_D$ that admits an appropriate extension onto the interior of the domain that we will also denote $g_D$ without loss of generality. The weak form to this problem can be derived by seeking $u \in \mathrm{H}{g_D}^1(\Omega)$ such that $$a(u, v):=\int{\Omega} \nabla u \cdot \nabla v=\int_{\Omega} f v=: L(v) \quad \forall v \in \mathrm{H}0^1(\Omega) .$$ An appropriate finite element space would be to define an interpolant, $I_h g_D$, of the Dirichlet data and set $$V{h, g_D}:=\left{v_h \in \mathbb{P}^1(\mathscr{T}):\left.v_h\right|{\partial \Omega}=I_h g_D\right},$$ where $\mathscr{T}$ denotes the triangulation. Notice that $V_h \not \subset \mathrm{H}{g_D}^1(\Omega)$ since $I_h g_D \neq g_D$ in general. As described in lectures, note also that we cannot compute $L(v)$ exactly in general. The finite element approximation is to find $u_h \in V_{h, g_D}$ such that
$$a\left(u_h, v_h\right)=L_h\left(v_h\right) \quad \forall v_h \in V_{h, 0},$$
where $L_h$ is an approximation to $L$.
Now let’s decompose the PDE solution and FE solution $u_0:=u-g_D$ and $u_{h, 0}:=u_h-I_h g_D$. Show that:
(1) The homogeneous parts of the error satisfy:
$\quad a\left(u_0-u_{h, 0}, v_h\right)=a\left(u-u_h, v_h\right)+a\left(I_h g_D-g_D, v_h\right) \quad \forall v_h \in V_{h, 0}$
(2) Let $v_h \in V_{h, 0}$ and set $w_h:=u_{h, 0}-v_h$. Show that:
$$\left|\nabla w_h\right|_{\mathrm{L}^2(\Omega)}^2 \leq a\left(u_0-v_h, w_h\right)-L\left(w_h\right)+L_h\left(w_h\right)+a\left(g_D-I_h g_D, w_h\right)$$

(3) Hence or otherwise show that
$$\left|\nabla w_h\right|_{\mathrm{L}^2(\Omega)} \leq C\left(\left|\nabla u_0-\nabla v_h\right|_{\mathrm{L}^2(\Omega)}+\frac{\left|L_h\left(w_h\right)-L\left(w_h\right)\right|}{\left|\nabla w_h\right|_{\mathrm{L}^2(\Omega)}}+\left|\nabla g_D-\nabla I_h g_D\right|_{\mathrm{L}^2(\Omega)}\right)$$
(4) Finally conclude the argument with the triangle inequality to show that
$$\left|\nabla u-\nabla u_h\right|_{L^2(\Omega)} \leq C\left(\inf {v_h \in V{h, 0}}\left|\nabla u_0-\nabla v_h\right|_{L^2(\Omega)}+\left|\nabla g_D-\nabla I_h g_D\right|_{L^2(\Omega)}+\sup {w_h \in V_h}\left|\frac{L_h\left(w_h\right)-L\left(w_h\right)}{\left|\nabla w_h\right|{L^2(\Omega)}}\right| \cdot\right)$$
(5) Relate this result to what we have studied in lectures on quadrature and best approximation to derive an explicit rate of convergence.
[25 marks]

Exercise 2.3: Adaptive finite elements ${<10$ pages $}$
In the lectures we showed the a posteriori error bound
$$\left|\nabla u-\nabla u_h\right|_{L^2(\Omega)} \leq C\left(\sum_{T \in \mathscr{T}} \eta_T^2\right)^{1 / 2},$$
with
$$\eta_T^2:=\left(h_T^2\left|\Delta u_h+f\right|_{\mathrm{L}^2(T)}^2+\frac{1}{2} \sum_{e \in \partial T} h_e\left|\llbracket \nabla u_h \rrbracket\right|_{\mathrm{L}^2(e)}^2\right) .$$
The code you have been provided has an implementation of this bound and uses it to locally design adaptive finite element schemes.

Carry out a study on the adaptive strategies implemented. These are explained in the accompanying document and the parameters are located in the init_data.m file. Study the problems from Exercises 2.1 and 2.2. This is an open ended problem. Some questions to think about are below, don’t try to answer all, focus on one/some and cover these fully.

• Can adaptive FEM achieve optimal convergence rates (measured in number of degrees of freedom, not $h)$ ?
• How do the adaptive parameters effect the approximation?

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