\begin{algorithmic}[1]
\Require $x_0\in K_0 \in\mathcal{K}_h(\Omega)$ and $x_1$ vertices of a 1-simplex $e$.
\Ensure Number of elements $N$, elements $\{K_0,..,K_{N-1}\}\in\mathcal{K}_h(\Omega)^N$.
\State $K\gets K_0$
\State $F_{\text{old}}\gets $ NULL
\State $K_{\text{old}}\gets $ NULL
\State $N \gets  1$
\State $E \gets \{K\}$
\While{$x_1\notin K$}
    \State  Find the isoparametric mapping $\phi_K:K_{\text{ref}}\mapsto K$ 
    \State  Find face $F\subset\partial K$ s.t. $F\neq F_{\text{old}}$ and $\phi_K^{-1}(e) \cap \phi_K^{-1}(F)\neq \emptyset$ 
    \State  $K\gets K\in\mathcal{K}_h(\Omega)$ s.t. $F\subset\partial K$ and $K\neq K_{\text{old}}$ ($K$ on the other side of face $F$)
    \State  $F_{\text{old}}\gets F$
    \State $N\gets N+1$
    \State $E \gets E \cup \{K\}$
\EndWhile
\end{algorithmic}

\begin{algorithmic}
[1]
\Require $x_0\in K_0 \in\mathcal{K}_h(\Omega)$ and $x_1$ vertices of a 1-simplex $e$.
\Ensure Number of elements $N$, elements $\{K_0,..,K_{N-1}\}\in\mathcal{K}_h(\Omega)^N$.
\State $K\gets K_0$
\State $F_{\text{old}}\gets $ NULL
\State $K_{\text{old}}\gets $ NULL
\State $N \gets 1$
\State $E \gets \{K\}$
\While{$x_1\notin K$}
\State Find the isoparametric mapping $\phi_K:K_{\text{ref}}\mapsto K$
\State Find face $F\subset\partial K$ s.t. $F\neq F_{\text{old}}$ and $\phi_K^{-1}(e) \cap \phi_K^{-1}(F)\neq \emptyset$
\State $K\gets K\in\mathcal{K}_h(\Omega)$ s.t. $F\subset\partial K$ and $K\neq K_{\text{old}}$ ($K$ on the other side of face $F$)
\State $F_{\text{old}}\gets F$
\State $N\gets N+1$
\State $E \gets E \cup \{K\}$
\EndWhile
\end{algorithmic}
{
    "title": "semi-lagrangian finite-element exterior calculus for incompressible   flows",
    "id": "2301.04923",
    "abstract": "we develop a mesh-based semi-lagrangian discretization of the time-dependent incompressible navier-stokes equations with free boundary conditions recast as a non-linear transport problem for a momentum 1-form. a linearly implicit fully discrete version of the scheme enjoys excellent stability properties in the vanishing viscosity limit and is applicable to inviscid incompressible euler flows. conservation of energy and helicity are enforced separately.",
    "categories": "math.na cs.na",
    "doi": "",
    "created": "2023-01-12",
    "updated": "2024-02-02",
    "authors": [
        "wouter tonnon",
        "ralf hiptmair"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2301.04923"
}
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