Due **Friday, January 10**.

This assignment is about the discretization of the Laplacian using FEM on triangular meshes, and the harmonic forms of the Hodge decomposition. Feel free to ask any questions by writing a blog post!

**1.** *( The stiffness matrix — 10 pts. ) *On a discrete triangular mesh \(M\), the stiffness matrix \(L\) is given by

\begin{align*}

L_{ij} = -\int_M d \varphi_{i}\wedge *d\varphi_j = -\sum_{t_{ijk}\succ e_{ij}}\langle\textrm{grad}\varphi_i,\textrm{grad}\varphi_j\rangle A_{ijk}

\end{align*}

where \(A_{ijk}\) is the area of triangle \(t_{ijk}\). Here we used the fact that the hat functions \(\varphi_i\), \(\varphi_j\) are piecewise linear and hence their gradients are piecewise constant on the triangles. We also used the fact that \(\textrm{grad}\varphi_i\) and \(\textrm{grad}\varphi_j\) have common support (the region on which the function is non-zero) only on the triangles incident to both \(i\) and \(j\).

(a) Show that the aspect ratio of a triangle can be expressed as the sum of the cotangents of the interior angles at its base, i.e.,

\begin{align*}

\frac{w}{h} = \cot\alpha + \cot\beta.

\end{align*}

(b) Show that the gradient of the hat function on triangle \(t_{ijk}\) is given by

\begin{align*}

\textrm{grad}\varphi_i = \frac{e_{jk}^{\bot}}{2A_{ijk}}

\end{align*}

where \(e^{\bot}_{jk}\) is the vector \(e_{jk}\) rotated \(90^\circ\) counterclockwise within \(t_{ijk}\).

(c) Show that for any hat function \(\varphi_i\) associated with vertex \(p_i\) of triangle \(t_{ijk}\), \[\langle\textrm{grad}\varphi_i,\textrm{grad}\varphi_i\rangle A_{ijk}=\tfrac{1}{2}(\cot\alpha+\cot\beta).\]

(d) Show that for the hat functions \(\varphi_i\) and \(\varphi_j\) associated with vertices \(p_i\) and \(p_j\) of triangle \(t_{ijk}\), we have

\begin{align*}

\langle\textrm{grad}\varphi_i,\textrm{grad}\varphi_j\rangle A_{ijk} = -\tfrac{1}{2}\cot\theta

\end{align*}

where \(\theta\) is the angle between the opposite edge vectors.

Putting all these facts together, we have the infamous *cotan formula*

\begin{align*}

(Lu)_i = \tfrac{1}{2}\sum_{e_{ij}\succ p_i}(\cot\alpha_{ij}+\cot\beta_{ij})(u_j-u_i).

\end{align*}

where \(\alpha_{ij}\) and \(\beta_{ij}\) are the angles of opposite vertices across from \(e_{ij}\) in the two adjacent triangles.

**2. ***( Harmonic forms on closed manifolds — 10 pts. ) * Show that, on a closed manifold \(M\) without boundary,

\begin{align*}

{\cal H}^k(M) = \left\{h\in\Omega^kM~\Big|~\Delta h = 0\right\},\quad\mbox{where \(\Delta = -d\delta – \delta d\).}

\end{align*}

That is, harmonic fields are the solutions to the Laplace’s equation, and vice versa. What happens if \(M\) has a boundary?