I have a bit of a naive question:
Why are there researchers in PDE theory, e.g. people who work on analysis and PDEs, when instead one can spend their research days building numerical solvers to get solutions that are as close to an exact answer as we want?
I could see how maybe 50 - 100 years ago, without computers / computing power, PDE researchers would be theorists and pure mathematicians, but why are there still pure PDE researchers today?
I know many of them win big awards / get lots of recognition, but I don't really understand what they do and why it's important, when we have so much computing power today.