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 -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 . Mimetic interpolation is particularly relevant in the context of vector and pseudo-vector fields as the method conserves line integrals and fluxes, respectively. Interpolation of integrated forms 
Let  be a differential -form, then mimetic interpolation is the linear combination

where  is the interpolation of , and the coefficients  represent the strengths of the field on grid element . Depending on ,  can be a node (), a cell edge (), a cell face () or a cell volume (). In the above, the  are the interpolating -forms, which are centered on  and decay away from  in a way similar to the tent functions. Examples of  are the Whitney forms for simplicial meshes in  dimensions. An important advantage of mimetic interpolation over other interpolation methods is that the field strengths  are scalars and thus coordinate system invariant. Interpolating forms 
In many cases it is desirable for the interpolating forms  to pick the field's strength on particular grid elements without interference from other . 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 (-forms) are zero on all nodes except on one, where the interpolating function is one. A similar construct can be applied to mimetic interpolation

That is, the integral of  is zero on all cell elements, except on  where the integral returns one. For  this amounts to  where  is a grid point. For  the integral is over  edges and hence the integral  is zero expect on edge . For  the integral is over faces and for  over cell volumes. Conservation properties 
Mimetic interpolation respects the properties of differential forms. In particular, Stokes' theorem

is satisfied with  denoting the interpolation of . Here,  is the exterior derivative,  is any manifold of dimensionality  and  is the boundary of . This confers to mimetic interpolation conservation properties, which are not generally shared by other interpolation methods. Commutativity between the interpolation operator and the exterior derivative 

To be mimetic, the interpolation must satisfy

where  is the interpolation operator of a -form, i.e. . In other words, the interpolation operators and the exterior derivatives commute. Note that different interpolation methods are required for each type of form (), . The above equation is all that is needed to satisfy Stokes' theorem for the interpolated form

Other calculus properties derive from the commutativity between interpolation and . For instance, ,

The last step gives zero since  when integrated over the boundary . Projection 

The interpolated  is often projected onto a target, -dimensional, oriented manifolds ,For  the target is a point, for  it is a line, for  an area, etc. Applications 
Many physical fields can be represented as -forms. When discretizing fields in numerical modeling, each type of -form often acquires its own staggering in accordance with numerical stability requirements, e.g. the need to prevent the checkerboard instability. This led to the development of the exterior finite element 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:

Example 

Consider quadrilateral cells in two dimensions with their node indexed  in the counterclockwise direction. Further, let  and  be the parametric coordinates of each cell (). Then

are the bilinear interpolating forms of  in the unit square (). The corresponding  edge interpolating forms are

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 ,

where  is the wedge product. We can verify that the above interpolating forms satisfy the mimetic conditions  and . Take for instance,

where , ,  and  are the field values evaluated at the corners of the quadrilateral in the unit square space. Likewise, we have

with , , being the 1-form projected onto edge . Note that  is also known as the pullback. If  is the map that parametrizes edge , , , then where the integration is performed in  space. Consider for instance edge   , then  with  and  denoting the start and points. For a general 1-form , one gets . References

Interpolation
Differential forms