Reduce algorithmic complexity of get_coupling_boundary_edges

[uekerman]

We currently do: https://github.com/precice/fenics-adapter/blob/e16d434345553f8c8cccf8d0804c6af94168e7c1/fenicsprecice/adapter_core.py#L213-L217

and within are_connected_by_edge: https://github.com/precice/fenics-adapter/blob/e16d434345553f8c8cccf8d0804c6af94168e7c1/fenicsprecice/adapter_core.py#L180-L184

which means O(N^4) :exploding_head: This makes volume coupling quasi infeasible. Should be possible in O(N).

[BenjaminRodenberg]

I will take care of this issue today. First some good news:

 for v1 in vertices.keys(): 
     for v2 in vertices.keys(): 
         if are_connected_by_edge(v1, v2): 
             vertices[v1] = v2 
             vertices[v2] = v1 

Is O(N^2) with N equal to the total number of vertices in the mesh. However,

 for edge1 in dolfin.edges(v1): 
     for edge2 in dolfin.edges(v2): 
         if edge1.index() == edge2.index():  # Vertices are connected by edge 
             return True 
 return False 

Only iterates over the edges of v1 times the edges of v2. Meaning O(M^2) with M << N.

In the end we arrive at a complexity of c*O(N^2), where c is a constant describing the number of edges per vertex ^ 2.