Algorithm 2.

Elemental Coupling Scheme

1:	Define the adaptive quadrature rule as a composite Gauss rule such that
$${\mathbb{A}}_q=\left\{\left({\mathit{X}}_q,w_q\right)\right\}\text{in which}\forall \mathit{X}\in {\text{Ω}}_0^{\text{s}},\underset{q}{\min }\Vert {\chi }_h(\mathit{X})-{\chi }_h\left({\mathit{X}}_q\right)\Vert \le C_{\mathbb{A}}\text{Δ}x$$ (68)
	in which typically $C_{\mathbb{A}}=1/2$
2:	procedure Velocity Interpolation $\left({\chi }_h,{\mathit{u}}_h,{\delta }_h\right)$	▹ Interpolate and project ${\mathit{u}}_h$ in the FE space
3:	Compute the components of ${\mathit{U}}^{\text{IB}}$, for each $\left({\mathit{X}}_q,w_q\right)\in {\mathbb{A}}_q$, as
$$U^{\text{IB},1}\left({\mathit{X}}_q\right)=\sum _{i,j,k}{u}_{i+\frac{1}{2},j,k}^1{\delta }_h\left({\mathit{x}}_{i+\frac{1}{2},j,k}-{\chi }_h\left({\mathit{X}}_q\right)\right)\text{Δ}{x}^3$$ (69a)
$$U^{\text{IB},2}\left({\mathit{X}}_q\right)=\sum _{i,j,k}{u}_{i,j+\frac{1}{2},k}^2{\delta }_h\left({\mathit{x}}_{i,j+\frac{1}{2},k}-{\chi }_h\left({\mathit{X}}_q\right)\right)\text{Δ}{x}^3$$ (69b)
$$U^{\text{IB},3}\left({\mathit{X}}_q\right)=\sum _{i,j,k}{u}_{i,j,k+\frac{1}{2}}^3{\delta }_h\left({\mathit{x}}_{i,j,k+\frac{1}{2}}-{\chi }_h\left({\mathit{X}}_q\right)\right)\text{Δ}{x}^3$$ (69c)
4:	Set up the projection right-hand side ${\overrightarrow{\mathit{L}}}^{\text{IB}}$ with entries
$$L_i^{\text{IB}}=\sum _{\left({\mathit{X}}_q,w_q\right)\in {\mathbb{A}}_q}{\mathit{U}}^{\text{IB}}\left({\mathit{X}}_q\right)\cdot {\varphi }_i\left({\mathit{X}}_q\right)w_q$$ (70)
5:	Solve $\mathbb{M}\overrightarrow{\mathit{U}}={\overrightarrow{L}}^{\text{IB}}$ for $\overrightarrow{\mathit{U}}$	▹ $\mathbb{M}$ is the standard mass matrix
6:	return $\overrightarrow{\mathit{U}}$	▹ vector of FE velocity coefficients
7:	end procedure
8:	procedure Force Spreading $\left({\chi }_h,{\delta }_h\right)$
9:	Calculate $\overrightarrow{\mathit{L}}$ with
$$L_i=-{\sum }_{\left({\mathit{X}}_q,w_q\right)\in {\mathbb{H}}_q}{\mathrm{\mathbb{P}}}^{\text{e}}\left({\mathit{X}}_q,t\right):{\nabla }_{\mathit{X}}{\varphi }_i\left({\mathit{X}}_q\right)w_q$$ (71)
10:	Solve $\mathbb{M}\overrightarrow{\mathit{F}}=\overrightarrow{\mathit{L}}$ for $\overrightarrow{\mathit{F}}$	▹ $\mathbb{M}$ is the standard mass matrix
11:	Define ${\mathit{F}}_h(\mathit{X})={\sum }_i{F}_i{\varphi }_i(\mathit{X})$
12:	Spread ${\mathit{F}}_h(\mathit{X})$ with into ${\mathit{f}}_h$ with
$$f_{i+\frac{1}{2},j,k}^1=\sum _{\left({\mathit{X}}_q,w_q\right)\in {\mathbb{A}}_q}{F}^1\left({\mathit{X}}_q\right){\delta }_h\left({\mathit{x}}_{i+\frac{1}{2},j,k}-{\chi }_h\left({\mathit{X}}_q\right)\right)w_q$$ (72a)
$$f_{i,j+\frac{1}{2},k}^2=\sum _{\left({\mathit{X}}_q,w_q\right)\in {\mathbb{A}}_q}{F}^2\left({\mathit{X}}_q\right){\delta }_h\left({\mathit{x}}_{i,j+\frac{1}{2},k}-{\chi }_h\left({\mathit{X}}_q\right)\right)w_q$$ (72b)
$$f_{i,j,k+\frac{1}{2}}^3=\sum _{\left({\mathit{X}}_q,w_q\right)\in {\mathbb{A}}_q}{F}^3\left({\mathit{X}}_q\right){\delta }_h\left({\mathit{x}}_{i,j,k+\frac{1}{2}}-{\chi }_h\left({\mathit{X}}_q\right)\right)w_q$$ (72c)
13:	return ${\mathit{f}}_h$	▹ Cartesian grid force representation
14:	end procedure