I would need a detailed mathematical derivation of mimetic finite difference operators in cartesian as well as curviliniear, in particular spherical coordinates. I also need to know how the pole problem in spherical coordinates can be tackled when using mimetic operators. Does anybody have a hint for me?
As far as I understand, the mimetic finite difference method is a method of spatial discretization of partial differential equation (in particular, in 3D). The key idea is about discretizing the gradient, divergence, rotation and Laplacian operators in such a way that well-known operator identities such as rot frad = 0, div rot = 0, div grad = Laplacian etc. are exactly fulfilled also for the discretized versions of the operators. Also, the method promises to conserve energy, mass, or zero-divergence during time-integrations of the partial differentieal equations, if I understand.
The following textbook seems to explain the method in-depth: https://www.amazon.de/Mimetic-Difference-Elliptic-Problems-English-ebook/dp/B0CX9J7LLP
But it seems to me, the application to curvilinear coordinates is not explained there. Therefore my question.
Thank you very much. Yesterday I found some articles by the same authors but with very little explanations, but in this article they do explain at least something. I will check this out.
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