The necessity for the application of numerical methods is best expressed by Paris and Hurd: “Most problems that can be solved formally (analytically) have been solved” [*]. In fact, until 1940s, most of electromagnetic problems that involve simplicity were solved by analytical methods, mainly by separation of variables and integral equation methods.
Still I see that many researchers trying to publish papers about analytical solutions to some simple problems that can't be considered as typical engineering problems. What is the point behind this research? maybe something I miss.
[*] Basic Electromagnetic Theory, D.T. Paris and F.K. Hurd, McGraw-Hill, New York, 1969, p. 166.