[HakikiCanakkaleli]

Hi

== 1 ==

The question is why Laplacian is considered as a single operator instead of a combination of the divergence and gradient operations despite these two mathematically the same.

== 2 ==

OpenFOAM programmer's guide Version 2.2.2, page 24, raises this point without giving any justifications:

The Laplacian is an operation that can be defined mathematically by a combination of the divergence and gradient operators by ∇2 ≡ ∇ • ∇. However, the Laplacian should be considered as a single operation that transforms a tensor field into another tensor field of the same rank, rather than a combination of two operations, one which raises the rank by 1 and one which reduces the rank by 1.

== 3 ==
Some made mention of this in the forum as follows:

Originally Posted by bigphil

...

Mathematically both expressions are the same but numerically the laplacian is calculated using a compact molecule (i.e. each face uses just the cell centre either side) whereas the second expression calculates the full gradient using all the surrounding face neighbour-neighbours and then takes the normal component.
...

== 4 ==

My guess is that changing the rank of the tensor of interest would lead to an arbitrary issue during the tensor transformation via Laplacian operator if Laplacian is implemented as a combination of the divergence and gradient operators.

== 5 ==

Many thanks for any answer.