Relations between number of cells, vertices and faces

For a two-dimensional triangular mesh, the number of cells $C$, the number of boundary faces $E_b$ and internal faces $E_i$ are related according to:

  $$E_b + 2 E_i = 3C$$ (8.1)

The total number of faces $E = E_i + E_b$. With $V$ the number of vertices and $H$ the number of holes ('islands'), we have the following Euler's relation for a triangulation:

  $$C + V - E = 1 - H$$ (8.2)

Usually, $E_b << E_i$ and the number of holes $H$ is negligibly small, so

  $$C \approx 2V\, , \quad E \approx 3V$$ (8.3)

There are approximately twice as many cells as vertices in a triangular mesh. Therefore, it is an optimal choice to locate the action density in vertices as the number of unknowns is minimal on a given grid. Concerning the time-consuming evaluation of the physical processes representing the wave energy generation, dissipation and redistribution, this allows SWAN to save a considerable amount of computing time.