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What are the "Pros and Cons" for these two methods of mesh-refinement? Both seem to be the prevalent methods.

I can naturally imagine that global red refinement is comparatively easy to implement and attractive for research in multigrid-methods, while NVB is used in AFEM-theory because some attractive formal properties that ease error analysis, and furthermore local adaption with NVB seems easier to implement than with Red-Green-Refinement.

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Here a most likely incomplete list. But maybe others can help extend it.

Bisection methods can be applied to simplicial meshes and have the advantage that they are always conforming.

Refinement into similar objects can be applied to triangles, quadrilaterals and hexahedra. To tetrahedra with some modifications. It allows for particularly simple setup of finite element multigrid methods, since the local embedding operators are always the same. It is very simple if combined with the technique of hanging nodes. And with p4est, there is a very efficient parallel representation for such meshes available.

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