Name

Definition

$\Omega$

Computational domain

${\Omega }_0$

Subdomain of $\Omega$

${\Omega }_1$

Subdomain of $\Omega$

$\Sigma$

Immersed boundary of the interface with $\Omega ={\Omega }_0\cup \Sigma \cup {\Omega }_1$

$n$

Normal unit outward vector to ${\Omega }_0$ on $\Sigma$

${\mathcal{P}}_b$

Model immersed boundary problem

${\mathcal{P}}_i$

Model immersed interface problem

$\epsilon$

Penalty parameter

$\chi$

Indicator function of ${\Omega }_1$

$\phi$

Value of the solution jump on $\Sigma$

$\psi$

Value of the flux jump on $\Sigma$

${\mathcal{N}}_0\mathrm{,}{\mathcal{N}}_1$

Set of index of the Eulerian interface points

${\mathcal{V}}_I$

Set of cell-centered finite volume

${{\mathcal{V}}^{\prime }}_I$

Set of dual finite volume

$\mathcal{I}$

Set of index of the intersection points between ${\sigma }_e$ and the faces ${\sigma }_l$

${\mathcal{L}}_f$

Set of index of the elements of the Lagrangian mesh

${\mathcal{L}}_v$

Set of index of the vertices of the Lagrangian mesh

${\sigma }_e$

Faces of the Eulerian dual mesh

${\sigma }_l$

Faces of the Lagrangian mesh

$x_{l\mathrm{,}i}$

Vertices of face ${\sigma }_l$ for $i=1,d$

${\Sigma }_h$

Piecewise linear approximation of $\Sigma$ such as ${\Sigma }_h=\left\{{\sigma }_l\in {\mathbb{P}}_1^{d-1}\mathrm{,}l\in {\mathcal{L}}_f\right\}$

$C\left(x_I\right)$

Phase ratio in the control volume centered in $x_I$

$d_I$

Unsigned distances between Eulerian points $x_I$ and the interface ${\Sigma }_h$