# Refinement in AMR

Assume I start with an 8x8 coarse mesh (see Fig 1) where the vertices (except boundary vertices) represent the unknown variable.

After iterative approximation - I find 2 vertices (unknowns) where the error exceeds a certain bound (see Figure 2 red dots).

Now I further refine the red dots (high error points) (see Figure 3 - only 1 red dot refinement shown).

My questions are:

1. Can I interpolate/prolongate the red X's (hanging nodes) from black X's of the coarser grid ? i.e. I don't need to solve PDE there.
2. Do the black X's and red X's become boundaries for green X's ? (I want to apply Finite Difference at green X's).
3. I believe refinement can be done to any level but generally a 2:1 balance is maintained to have a single hanging node per edge (in 2-D). Do we refine multiple times before updating or just once ?
4. I will be using a quad tree with each tree node containing 4 vertices (with ownership status) - for a vertex centred finite difference implementation but it will require querying adjacent neighbours (H. Samet, 1982 paper) which looks in-efficient. Any suggestions on this ?