I work in a discipline in which a scientist ,Greenwood, has introduced a function to describe a relationship. Some times in the articles which I read I see some authors call the function as "Greenwood's function" others "the Greenwood function". I have seen this issue elsewhere like "the Obama administration"
Which one is correct and what is the rule for allocating or relating things to people? Is it different from the possession rules?
There are no solid rules about this. You have to judge by context.
Within a single academic paper concerning the function I think "Greenwood's function" and "the Greenwood function" are identical. But in the wider literature the possibility of ambiguity grows because Greenwood may have created or contributed to more than one function.
In your "Obama administration" example, some speakers may be content to consider his whole eight years as one administration, while more finicky people prefer to focus on the differences between his first term and his second.