Monthly Archives: July 2016

Wave equation Let $M$ be a surface and $f : \mathbb{R} \times M \rightarrow \mathbb{R}$ a function. The wave equation is given by: \[\ddot{f} = \Delta f.\] After the space discretization we have $f:\mathbb{R} \times V \rightarrow \mathbb{R},$ $p:= \left(\begin{matrix} … Continue reading →

Let $\Omega \subset \mathbb{R}^n$ be a domain, we will consider $\Omega$ to be the “space” and functions: \begin{align} f : \mathbb{R} \times \Omega \rightarrow \mathbb{R}, \\ (t,x) \mapsto f(t,x),\end{align} will be view as time dependent functions \begin{align} & f_t : … Continue reading →

Let $M= \left(V, E, F \right)$ be an oriented triangulated surface without boundary and $p:V \rightarrow \mathbb{R}^3$ a realization. In earlier lectures we considered the space of piecewise linear functions on $M$: \[W_{PL}:=\left\{ \tilde{f} :M\rightarrow \mathbb{R} \, \big \vert \,\left. … Continue reading →

Let $\Sigma = (V,E,F)$ be a triangulated surface (with boundary). A realization of the surface in $\mathbb{R}^3$ is given by a map $p:V \rightarrow \mathbb{R}^3$ such that $p_i,p_j,p_k$ form a non degenerated triangle in $\mathbb{R}^3$ for all $\{i,j,k\} \in \Sigma$, … Continue reading →

Let $M = (V,E,F)$  be a triangulated domain in the plane where $V$ denotes the set of vertices, $E$ the set of edges and $F$ the set of triangles.  We consider the set of functions on the vertices  : \begin{align*} … Continue reading →

Let $M$ be a triangulated domain with the functionspace: \[W_{PL}:=\bigl\{f:M\rightarrow\mathbb{R}\,\bigl\vert\bigr.\,\,\,\left. f\right|_{T_{\sigma}} \mbox{ is affine for all } \sigma \in \Sigma_2 \bigr\}.\] On the interior of each triangle $T_{\sigma}$ in $M$ the gradient of a function $g \in W$ is well … Continue reading →

We want to derive a discrete version of the laplace operator defined on triangulated surfaces. At first we will define what a triangulated surface with and without  boundary is,  and consider triangulated domains of $\mathbb{R}^2$ as an important example. Let … Continue reading →

Let $M \subset \mathbb{R}^2$ be a domain with smooth boundary $\partial M$ and outpointing normal vector field $N$. For a smooth function $f \in C^{\infty}(M,\mathbb{R})$ the gradient vector field $\mbox{grad} \, f :M \rightarrow \mathbb{R}^2$ is defined as : \[ … Continue reading →