Mimetic interpolation

In mathematics, mimetic interpolation is a method for interpolating differential forms. In contrast to other interpolation methods, which estimate a field at a location given its values on neighboring points, mimetic interpolation estimates the field's ${\displaystyle k}$-form given the field's projection on neighboring grid elements. The grid elements can be grid points as well as cell edges or faces, depending on ${\displaystyle k=0,1,2,\cdots }$.

Mimetic interpolation is particularly relevant in the context of vector and pseudo-vector fields as the method conserves line integrals and fluxes, respectively.

Let ${\displaystyle \omega ^{k}}$ be a differential ${\displaystyle k}$-form, then mimetic interpolation is the linear combination

$${\displaystyle {\bar {\omega }}^{k}=\sum _{i}\left(\int _{M_{i}}\omega ^{k}\right)\phi _{i}^{k}}$$

where ${\displaystyle {\bar {\omega }}^{k}}$ is the interpolation of ${\displaystyle \omega ^{k}}$, and the coefficients ${\displaystyle \int _{M_{i}}\omega ^{k}}$ represent the strengths of the field on grid element ${\displaystyle M_{i}}$. Depending on ${\displaystyle k}$, ${\displaystyle M_{i}}$ can be a node (${\displaystyle k=0}$), a cell edge (${\displaystyle k=1}$), a cell face (${\displaystyle k=2}$) or a cell volume (${\displaystyle k=3}$). In the above, the ${\displaystyle \phi _{i}^{k}}$ are the interpolating ${\displaystyle k}$-forms, which are centered on ${\displaystyle M_{i}}$ and decay away from ${\displaystyle M_{i}}$ in a way similar to the tent functions. Examples of ${\displaystyle \phi _{i}^{k}}$ are the Whitney forms^[1][2] for simplicial meshes in ${\displaystyle n}$ dimensions.

An important advantage of mimetic interpolation over other interpolation methods is that the field strengths ${\displaystyle \int _{M_{i}}\omega ^{k}}$ are scalars and thus coordinate system invariant.

In many cases it is desirable for the interpolating forms ${\displaystyle \phi _{i}^{k}}$ to pick the field's strength on particular grid elements without interference from other ${\displaystyle \phi _{j}^{k}}$. This allows one to assign field values to specific grid elements, which can then be interpolated in-between. A common case is linear interpolation for which the interpolating functions (${\displaystyle 0}$-forms) are zero on all nodes except on one, where the interpolating function is one. A similar construct can be applied to mimetic interpolation

${\displaystyle \int _{M_{j}}\phi _{i}^{k}=\delta _{ij}.}$

That is, the integral of ${\displaystyle \phi _{i}^{k}}$ is zero on all cell elements, except on ${\displaystyle M_{i}}$ where the integral returns one. For ${\displaystyle k=0}$ this amounts to ${\displaystyle \phi _{i}^{0}(x_{j})=\delta _{ij}}$ where ${\displaystyle x_{j}}$ is a grid point. For ${\displaystyle k=1}$ the integral is over edges and hence the integral ${\displaystyle \int _{M_{j}}\phi _{i}^{1}}$ is zero expect on edge ${\displaystyle M_{i}}$. For ${\displaystyle k=2}$ the integral is over faces and for ${\displaystyle k=3}$ over cell volumes.

Mimetic interpolation respects the properties of differential forms. In particular, Stokes' theorem

$${\displaystyle \int _{M}{\overline {d\omega ^{k}}}=\int _{\partial M}{\bar {\omega }}^{k}}$$

is satisfied with ${\displaystyle {\overline {d\omega ^{k}}}}$ denoting the interpolation of ${\displaystyle d\omega ^{k}}$. Here, ${\displaystyle d}$ is the exterior derivative, ${\displaystyle M}$ is any manifold of dimensionality ${\displaystyle k}$ and ${\displaystyle \partial M}$ is the boundary of ${\displaystyle M}$. This confers to mimetic interpolation conservation properties, which are not generally shared by other interpolation methods.

To be mimetic, the interpolation must satisfy

$${\displaystyle I_{k+1}(d\omega ^{k})=d(I_{k}\omega ^{k})}$$

where ${\displaystyle I_{k}}$ is the interpolation operator of a ${\displaystyle k}$-form, i.e. ${\displaystyle {\bar {\omega }}^{k}=I_{k}\omega ^{k}}$. In other words, the interpolation operators and the exterior derivatives commute.^[3] Note that different interpolation methods are required for each type of form (${\displaystyle k=0,1,\cdots }$), ${\displaystyle I_{k+1}\neq I_{k}}$. The above equation is all that is needed to satisfy Stokes' theorem for the interpolated form

$${\displaystyle \int _{M}{\overline {d\omega ^{k}}}=\int _{M}I_{k+1}(d\omega ^{k})=\int _{M}d(I_{k}\omega ^{k})=\int _{\partial M}I_{k}\omega ^{k}=\int _{\partial M}{\bar {\omega }}^{k}.}$$

Other calculus properties derive from the commutativity between interpolation and ${\displaystyle d}$. For instance, ${\displaystyle d^{2}\omega =0}$,

$${\displaystyle \int _{M}I_{k+2}(d^{2}\omega ^{k})=\int _{M}d(I_{k+1}d\omega ^{k})=\int _{\partial M}I_{k+1}d\omega ^{k}=\int _{\partial M}d(I_{k}\omega ^{k})=0.}$$

The last step gives zero since ${\displaystyle \int _{\partial M}d(\cdot )=0}$ when integrated over the boundary ${\displaystyle \partial M}$.

The interpolated ${\displaystyle {\bar {\omega }}^{k}}$ is often projected onto a target, ${\displaystyle k}$-dimensional, oriented manifolds ${\displaystyle T}$,$${\displaystyle \int _{T}{\bar {\omega }}^{k}=\sum _{i}\left(\int _{M_{i}}\omega ^{k}\right)\left(\int _{T}\phi _{i}^{k}\right).}$$For ${\displaystyle k=0}$ the target is a point, for ${\displaystyle k=1}$ it is a line, for ${\displaystyle k=2}$ an area, etc.

Many physical fields can be represented as ${\displaystyle k}$-forms. When discretizing fields in numerical modeling, each type of ${\displaystyle k}$-form often acquires its own staggering in accordance with numerical stability requirements, e.g. the need to prevent the checkerboard instability.^[4] This led to the development of the exterior finite element^[5] and discrete exterior calculus methods, both of which rely on a field discretization that are compatible with the field type.

The table below lists some examples of physical fields, their type, their corresponding form and interpolation method, as well as software that can be leveraged to interpolate, remap or regrid the field:

Consider quadrilateral cells in two dimensions with their node indexed ${\displaystyle i=0,1,2,3}$ in the counterclockwise direction. Further, let ${\displaystyle \xi _{1}}$ and ${\displaystyle \xi _{2}}$ be the parametric coordinates of each cell (${\displaystyle 0\leq \xi _{1},\xi _{2}\leq 1}$). Then$${\displaystyle {\begin{aligned}\phi _{0}^{0}&=(1-\xi _{1})(1-\xi _{2})\\[0.6ex]\phi _{1}^{0}&=\xi _{1}(1-\xi _{2})\\[0.6ex]\phi _{2}^{0}&=\xi _{1}\xi _{2}\\[0.6ex]\phi _{3}^{0}&=(1-\xi _{1})\xi _{2}.\end{aligned}}}$$

are the bilinear interpolating forms of ${\displaystyle I_{0}}$ in the unit square (${\displaystyle \xi _{1},\xi _{2}}$). The corresponding ${\displaystyle I_{1}}$ edge interpolating forms^[9][10] are

$${\displaystyle {\begin{aligned}\phi _{0}^{1}&=(1-\xi _{2})d\xi _{1}\\[0.6ex]\phi _{1}^{1}&=\xi _{1}d\xi _{2}\\[0.6ex]\phi _{2}^{1}&=\xi _{2}d\xi _{1}\\[0.6ex]\phi _{3}^{1}&=(1-\xi _{1})d\xi _{2},\end{aligned}}}$$

were we assumed the edges to be indexed in counterclockwise direction and with the edges pointing to the east and north. At lowest order, there is only one interpolating form for ${\displaystyle I_{2}}$,

$${\displaystyle \phi _{0}^{2}=d\xi _{1}\wedge d\xi _{2},}$$

where ${\displaystyle \wedge }$ is the wedge product.

We can verify that the above interpolating forms satisfy the mimetic conditions ${\displaystyle I_{1}d\omega ^{0}=d(I_{0}\omega ^{0})}$ and ${\displaystyle d(I_{1}\omega ^{1})=I_{2}d\omega ^{1}}$. Take for instance,

$${\displaystyle {\begin{aligned}d(I_{0}\omega ^{0})&=d\left(f_{0}(1-\xi _{1})(1-\xi _{2})+f_{1}\xi _{1}(1-\xi _{2})+f_{2}\xi _{1}\xi _{2}+f_{3}(1-\xi _{1})\xi _{2}\right)\\[0.6ex]&=(f_{1}-f_{0})(1-\xi _{2})d\xi _{1}+(f_{2}-f_{1})\xi _{1}d\xi _{2}+(f_{2}-f_{3})\xi _{2}d\xi _{1}+(f_{3}-f_{0})(1-\xi _{1})d\xi _{2}\\[0.6ex]&=(f_{1}-f_{0})\phi _{0}^{1}+(f_{2}-f_{1})\phi _{1}^{1}+(f_{2}-f_{3})\phi _{2}^{1}+(f_{3}-f_{0})\phi _{3}^{1}\\[0.6ex]&=I_{1}d\omega ^{0}\end{aligned}}}$$where ${\displaystyle f_{0}=\omega ^{0}(0,0)}$, ${\displaystyle f_{1}=\omega ^{0}(1,0)}$, ${\displaystyle f_{2}=\omega ^{0}(1,1)}$ and ${\displaystyle f_{3}=\omega ^{0}(0,1)}$ are the field values evaluated at the corners of the quadrilateral in the unit square space. Likewise, we have

$${\displaystyle {\begin{aligned}d(I_{1}\omega ^{1})&=d(g_{0}(1-\xi _{2})d\xi _{1}+g_{1}\xi _{1}d\xi _{2}+g_{2}\xi _{2}d\xi _{1}+g_{3}(1-\xi _{1})d\xi _{2})\\[0.6ex]&=(g_{0}+g_{1}-g_{2}-g_{3})d\xi _{1}\wedge d\xi _{2}\\[0.6ex]&=I_{2}d\omega ^{1}\end{aligned}}}$$ with ${\displaystyle g_{i}=\int _{M_{i}}\omega ^{1}}$, ${\displaystyle i\in \{0,1,2,3\}}$, being the 1-form ${\displaystyle \omega ^{1}}$projected onto edge ${\displaystyle i}$. Note that ${\displaystyle g_{i}}$ is also known as the pullback. If ${\displaystyle F:\mathbb {R} \rightarrow \mathbb {R} ^{2}}$ is the map that parametrizes edge ${\displaystyle i}$, ${\displaystyle x=F_{i}(t)}$, ${\displaystyle 0\leq t\leq 1}$, then ${\displaystyle g_{i}=\int {F_{i}}^{*}\omega ^{1}}$where the integration is performed in ${\displaystyle t}$ space. Consider for instance edge ${\displaystyle i=2}$ , then ${\displaystyle F_{2}(t)=x_{3}+t(x_{2}-x_{3})}$ with ${\displaystyle x_{2}}$ and ${\displaystyle x_{3}}$ denoting the start and points. For a general 1-form ${\displaystyle \omega ^{1}=a(x)d\xi _{1}+b(x)d\xi _{2}}$, one gets ${\displaystyle g_{2}=\int _{0}^{1}\left(a(t){\frac {\partial \xi _{1}}{\partial t}}+b(t){\frac {\partial \xi _{2}}{\partial t}}\right)dt}$.