In numerical simulations two methods are broadly used: Finite Element Method (FEM) and Finite Difference Method, and Finite Volume Method (there are also some additional methods not considered here). All three methods allow us to find approximate numerical solutions to systems of partial differential equations by reduction of the problems to a system of algebraic equations (with the unkwons representing the values of mappings at specific points).
For thermomechanical problems FEM is broadly used, and probably is dominant. For electromagnetic problems FEM and FDM are broadly used. In some areas of physics (e.g. Numerical Relativity) FDM is almost exclusively used.
Although from a mathematical point of view, at least for linear problems, all three can be applied to the same problem, it is not clear to me why in certain areas one is preferred over the other, my impression is that the reasons may not be mathematical, but I would like to know the opinion of someone mathematically very knowledgeable about what mathematical advantages one method or the other may bring depending on the nature of the problem.