Editing Interpolation (section)

=== Mimetic interpolation ===
{{Main|Mimetic interpolation}}
Depending on the underlying discretisation of fields, different interpolants may be required. In contrast to other interpolation methods, which estimate functions on target points, mimetic interpolation evaluates the integral of fields on target lines, areas or volumes, depending on the type of field (scalar, vector, pseudo-vector or pseudo-scalar).

A key feature of mimetic interpolation is that [[vector calculus identities]] are satisfied, including [[Stokes' theorem]] and the [[divergence theorem]]. As a result, mimetic interpolation conserves line, area and volume integrals.<ref>{{Cite journal |last1=Pletzer |first1=Alexander |last2=Hayek |first2=Wolfgang |date=2019-01-01 |title=Mimetic Interpolation of Vector Fields on Arakawa C/D Grids |url=https://journals.ametsoc.org/view/journals/mwre/147/1/mwr-d-18-0146.1.xml |journal=Monthly Weather Review |language=EN |volume=147 |issue=1 |pages=3–16 |doi=10.1175/MWR-D-18-0146.1 |bibcode=2019MWRv..147....3P |s2cid=125214770 |issn=1520-0493 |access-date=2022-06-07 |archive-date=2022-06-07 |archive-url=https://web.archive.org/web/20220607041035/https://journals.ametsoc.org/view/journals/mwre/147/1/mwr-d-18-0146.1.xml |url-status=live }}</ref> Conservation of line integrals might be desirable when interpolating the [[electric field]], for instance, since the line integral gives the [[electric potential]] difference at the endpoints of the integration path.<ref>{{Citation |last1=Stern |first1=Ari |title=Geometric Computational Electrodynamics with Variational Integrators and Discrete Differential Forms |date=2015 |url=http://link.springer.com/10.1007/978-1-4939-2441-7_19 |work=Geometry, Mechanics, and Dynamics |volume=73 |pages=437–475 |editor-last=Chang |editor-first=Dong Eui |place=New York, NY |publisher=Springer New York |doi=10.1007/978-1-4939-2441-7_19 |isbn=978-1-4939-2440-0 |access-date=2022-06-15 |last2=Tong |first2=Yiying |last3=Desbrun |first3=Mathieu |last4=Marsden |first4=Jerrold E. |series=Fields Institute Communications |s2cid=15194760 |editor2-last=Holm |editor2-first=Darryl D. |editor3-last=Patrick |editor3-first=George |editor4-last=Ratiu |editor4-first=Tudor|arxiv=0707.4470 }}</ref> Mimetic interpolation ensures that the error of estimating the line integral of an electric field is the same as the error obtained by interpolating the potential at the end points of the integration path, regardless of the length of the integration path. 

[[Linear interpolation|Linear]], [[Bilinear interpolation|bilinear]] and [[trilinear interpolation]] are also considered mimetic, even if it is the field values that are conserved (not the integral of the field). Apart from linear interpolation, area weighted interpolation can be considered one of the first mimetic interpolation methods to have been developed.<ref>{{Cite journal |last=Jones |first=Philip |title=First- and Second-Order Conservative Remapping Schemes for Grids in Spherical Coordinates |journal=Monthly Weather Review |year=1999 |volume=127 |issue=9 |pages=2204–2210|doi=10.1175/1520-0493(1999)127<2204:FASOCR>2.0.CO;2 |bibcode=1999MWRv..127.2204J |s2cid=122744293 |doi-access=free }}</ref>