What are the advantages and disadvantages of analytical and semi-analytical methods over numerical methods like FEM and FDM?
I am looking for detailed advantages and disadvantages.
Advantage:
You have exact answer with analytical method, and you can produce exact answer most of the time with semi-analytical method (one way analytical but does not produce results in terms of elementary functions so you need to numerically evaluate results to produce them, like Bessel Function, Error Function, Beta Function, Arey Function ....) You can use them for strong proofs, and checking results and new theories
Disadvantage:
The analytical and semi-analytical solution only exist in limited setups like, low dimensions, symmetric cases, special cases of BC or IC, Homogeneous coefficients, linear cases, etc... . Therefore you have to use "Numerical" solutions in general, it can be FDM, FEM, BEM, SPH, FEM, Spectral Method, Analytical Element Method, ...
Hope it helps,
Kaveh
In general, closed and exact solutions are possible only for simplified physical models that allow us to disregard (under suitable assumptions) some terms in the PDE. Therefore, from a side you have the advantage of an analytical solution, on the other side the physical validity of the solution is constrained by several hypotheses.
Numerical solutions generally works in a more general conditions of physical validity, the PDE are more complex and the advantage is the the solution has a more general validity. The great issue is in the final quality of the numerical solution, it can appear quite reasonable even if it is totally wrong!
However complicated a problem you are solving, which does not admit exact solution and you are forced to use numerical method, please do not ignore the fact that you MUST calibrate your adopted methods with the help of simple analytical solution cases. Thus, these approaches are complimentary in nature and is not an either/or issue. This is the only way to make rapid progress!
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