Is any of today's applied mathematics research in academia still done primarily on pencil and paper?

I am an applied mathematician and I can't work without paper and pencil and in most cases I can't get things done without a computer.

My computer can't tell me in advance if my algorithm will converge, at what rate and what the error will be. My computer can't figure out how to discretize some new problem (unless I tell him how). My computer (up to now) never had a great idea.

On the other hand, I can't invert matrices with pen and paper and I even can multiply a large matrix like that. In fact I can't even store most data I use on paper... (and also access rates for data on paper are pretty slow).

tl;dr I basically always use paper, pen and computer.

In addition to @Dirk's excellent answer, which I wholeheartedly concur with, I will also note that many types of applied mathematical research are done entirely without aid of a computer.

Consider, for example, this well-cited paper on Brewer's conjecture, which is all mathematical reasoning about the nature of distributed computations. At the more theoretical side of any applied field, you can find such papers, where the conclusions are in the symbolic mathematics that is still best carried out by hand, most typically on paper, whiteboard, or blackboard in its crucial early stages.

Of course, basically nobody submits physical paper to journals any more, so I guess a computer does get involved eventually...