Both Uses

notwithstanding
in
The Pit and the Pendulum
(Edited)

Such a supposition, notwithstanding what we read in fiction, is altogether inconsistent with real existence;—but where and in what state was I?

notwithstanding = in spite of; or in spite of the thing just mentioned (used to connect contrasting ideas)
Notwithstanding terrifically wide sweep (some thirty feet or more) and the its hissing vigor of its descent, sufficient to sunder these very walls of iron, still the fraying of my robe would be all that, for several minutes, it would accomplish.
*

notwithstanding = despite

Definitions:

(1) (notwithstanding)
used to show that something remains true even though something else seems to go against it

Most commonly, you can substitute "in spite of" for notwithstanding. But see the first exemplary sample sentences for other common word substitutions.
(2) (meaning too rare to warrant focus)