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Consider solving a differential equation system for 1D, 2D, or 3D.

It involves various input and output "field" variables, which, more often than not, correspond to various physical quantities relevant to the problem.

The typical discretization step is to agree on a mesh, and then solve the system. We might refine the mesh in some areas of the spatial-domain (1D, 2D, or 3D) to increase resolution.

However, each "field" has its own high-resolution requiring regions, which are, in general, different from every other field. E.g., considering 1D only, a sin(x) function is relatively flat around pi/2, but steep around 0, whereas cos(x) is flat around 0, but step around pi/2. That means a non-uniform mesh optimized for a sin 1D field is not optimal for a cos field, and vice versa. This means we need a mesh for sin, and another mesh for cos, if we are to do things in the most optimized way.

Despite spending many years in the area of scientific computing, I'm surprised that I've never come across such a discussion that highlights the need for "multi-mesh numerics". Granted it has its own problems, e.g., the proliferation of meshes for intermediate results, computational cost of conversion, and so on, but academically, it's a direction worth pursuing.

So my question is, can anyone help me point to a body of literature that covers this topic? Or is there a different term used for this that I'm not familiar with?

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  • $\begingroup$ Well, if there is no coupling you can solve them independently on different meshes, otherwise the resolution can play a significant role, at least where the coupling takes place (e.g. for integrating the source/boundary terms). If this coupling is not strong, perhaps very localized or of a different order, then it may be worthwhile to consider some tricks which are usually found under the terms "multiscale" methods, "splitting" methods and the like. I think however that multiple independent meshes come with high logistic overhead and I cannot think of too many situations where they make sense. $\endgroup$ – Christian Waluga Mar 1 '17 at 6:42
  • $\begingroup$ It might be interesting for you to search for distributed Lagrange multiplier methods. For example, ac.els-cdn.com/S0301932298000482/… and for Chimera method. I am not very familiar with these topics so if you find any good overviews please share the links here. $\endgroup$ – VorKir Mar 2 '17 at 20:43

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