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 $f:\mathbb{R} \times V \rightarrow \mathbb{R},$ $p:= \left(\begin{matrix} … Continue reading Posted in Lecture | Comments Off on Wave- and heat-equation on surfaces Partial differential equations involving time Let$\Omega \subset \mathbb{R}^n$be a domain, we will consider$\Omegato 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 Posted in Lecture | Comments Off on Partial differential equations involving time Laplace operator 2 LetM= \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 Posted in Lecture | Comments Off on Laplace operator 2 Triangulated surfaces with metric and the Plateau problem 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 Posted in Lecture | Comments Off on Triangulated surfaces with metric and the Plateau problem Dirichlet energy 2 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 Posted in Lecture | Comments Off on Dirichlet energy 2 Gradient and Dirichlet energy on triangulated domains. 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 triangleT_{\sigma}$in$M$the gradient of a function$g \in W$is well … Continue reading Posted in Lecture | Comments Off on Gradient and Dirichlet energy on triangulated domains. Triangulated surfaces and domains We want to derive a discrete version of the laplace operator defined on triangulated surfaces. At first we will define what a triangulated surface with and without boundary is, and consider triangulated domains of$\mathbb{R}^2$as an important example. Let … Continue reading Posted in Lecture | Comments Off on Triangulated surfaces and domains Laplace operator 1 Let$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

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