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 k {\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 k = 0 , 1 , 2 , ⋯ {\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.
Interpolation of integrated forms Let ω k {\displaystyle \omega ^{k}} be a differential k {\displaystyle k} -form, then mimetic interpolation is the linear combination
ω ¯ k = ∑ i ( ∫ M i ω k ) ϕ i k {\displaystyle {\bar {\omega }}^{k}=\sum _{i}\left(\int _{M_{i}}\omega ^{k}\right)\phi _{i}^{k}}
where ω ¯ k {\displaystyle {\bar {\omega }}^{k}} is the interpolation of ω k {\displaystyle \omega ^{k}} , and the coefficients ∫ M i ω k {\displaystyle \int _{M_{i}}\omega ^{k}} represent the strengths of the field on grid element M i {\displaystyle M_{i}} . Depending on k {\displaystyle k} , M i {\displaystyle M_{i}} can be a node ( k = 0 {\displaystyle k=0} ), a cell edge ( k = 1 {\displaystyle k=1} ), a cell face ( k = 2 {\displaystyle k=2} ) or a cell volume ( k = 3 {\displaystyle k=3} ). In the above, the ϕ i k {\displaystyle \phi _{i}^{k}} are the interpolating k {\displaystyle k} -forms, which are centered on M i {\displaystyle M_{i}} and decay away from M i {\displaystyle M_{i}} in a way similar to the tent functions. Examples of ϕ i k {\displaystyle \phi _{i}^{k}} are the Whitney forms for simplicial meshes in n {\displaystyle n} dimensions. An important advantage of mimetic interpolation over other interpolation methods is that the field strengths ∫ M i ω k {\displaystyle \int _{M_{i}}\omega ^{k}} are scalars and thus coordinate system invariant.
Interpolating forms In many cases, it is desirable for the interpolating forms ϕ i k {\displaystyle \phi _{i}^{k}} to pick the field's strength on particular grid elements without interference from other ϕ j k {\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 ( 0 {\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
∫ M j ϕ i k = δ i j . {\displaystyle \int _{M_{j}}\phi _{i}^{k}=\delta _{ij}.}
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