Capitalisation of "Section" and "Chapter" in a Ph.D. Thesis

"In Chapter 3, it was shown that..."

This seems correct. "Chapter 3" is the name of the third chapter. Names are capitalised.

"In the previous Section, a method was presented to..."

This seems wrong. "Section" is not referring to the previous section by name, therefore no capital.

"The graph in Figure 3 shows..."

Correct. Same as the first example.

So the rule (I use) is, if it is a proper name, then use a capital. This means, if it is of the form "Section $n$", where $n$ is a number, then it needs a capital.

It is a question of style. The most accepted custom is that given by Dave: you capitalize logical divisions if you refer to them by number.

However, I've never believed that there is any real logic behind that rule, other than emphasis. Identifying things by a number doesn't make them proper nouns: as an example, you don't commonly capitalize “page” as “see Page 10”…

A search on Google Scholar reveals that both the forms

in chapter/section 3

and

in Chapter/Section 3

exist in published scientific articles.

For "chapter" the capitalised version seems to be a little more common. For "section" the capitalised version is much more common.