Say I need to solve a PDE numerically. Depending on the problem and the numerical method chosen, I can usually see some issues coming: implementation issues (e.g. boundary conditions, parallelization), discretization issues (e.g. the stabilization of finite elements for hyperbolic problems is not funny while finite volumes "just work", the "opposite" for elliptic problems), time integration...
I usually base the initial choice of the discretization on these "intuitions" only. Still, I would love to base them on facts and proof regarding the question: how much computational effort would I have to put to solve my problem to a desired solution error (i.e. dollars/error)?
- Is there any resource I could use to weight the suitability of a method with this criterion?
Although one could compute the complexity of the methods one still would need a priori error estimators that, if they exist at all, would have different "sharpness", depending on the method. Is there a non-empirical way of finding this out? If testing all possible methods cannot be avoided, how would one go on about it? There are standard numerical testcases for most fields, would they be meaningful? Are there any public databases comparing implementations for these testcases?
Disclaimer: This should not be the only criterion, implementing finite difference for complex boundaries is "doable" but a pain. Still, is it worth it to go through that pain? If the cost of using FD would be much lower than the alternatives, for some people, it might.