Can FDM solve any type of PDE same as FEM?

[zoltrix]

hello

aside from some constraints such as an irregular integration domain, can FDM solve any type of PDE same as FEM ?

 

[jedishrfu]

To avoid any possible confusion here, what do you mean by FDM and FEM?

I’m guessing PDE is for partial differential equations.

 

[Delta2]

I think FDM is for Finite Difference Method, and FEM for Finite Element Method. Both methods for numerical solutions to Differential Equations.

 

[bigfooted]

If you work out the discretized (Galerkin) finite element scheme with linear basis functions for say the Laplace equation on a single cell of an equidistant quadrilateral mesh, how does this compare to for example the central difference scheme for the same cell? If you write it out for both methods you will see the differences and similarities. To be complete, you can also write out the resulting finite volume scheme for comparison.

Note that there are many finite element methods and many finite difference schemes. A specific Finite Element Method is limited in the types of equations that can be solved by it, just as a specific Finite Difference Method is limited in the types of equations that can be solved with it.

 

[zoltrix]

so, do you mean that it does not exist an universal FDM scheme which fits all types of PDE ?
same as the Runge Kutta method, for example, which is applicable to all types of ODE
is it just a matter of accuracy or some PDE schemes may be unstable for some type of PDE ?
as far as I know all FEM schemes yield results even though some are more accurate or faster than others depending upon the shape of the integration domain

 

[Frabjous]

so, do you mean that it does not exist an universal FDM scheme which fits all types of PDE ?

as far as I know all FEM schemes yield results even though some are more accurate or faster than others depending upon the shape of the integration domain

You are holding FDM to a much higher standard than FEM.

Asking in full generality is probably hindering the response to this question because people are worried about edge cases. Be more specific about what you are concerned about.

 

[zoltrix]

consider linear equation of second order
a common classification is : elliptic-parabolic-hyperbolic

is FDM suitable for all these types of PDE ?

 

[Frabjous]

Yes

 

[zoltrix]

Is it also possible to write in c or python a general function :

solver(eqn,a,b,c,d...)

or at least a specific "solver" for each type of PDE's whereas "eqn" is the generic partial linear differential equation of second order and a,b,c,d... its parameters ?
if so , can you suggest a book or a website ?

 

[bigfooted]

I recently did a project with a student in python where the online works of Langtangen were used a lot:
http://hplgit.github.io/

https://hplgit.github.io/fdm-book/doc/web/index.html
https://hplgit.github.io/fdm-book/doc/pub/book/html/decay-book.html

http://hplgit.github.io/num-methods-for-PDEs/doc/pub/index.html
http://hplgit.github.io/num-methods-for-PDEs/doc/pub/nonlin/html/nonlin.html

 

[zoltrix]

Thanks bigfooted

it is exactly what I was looking for

 

[Chestermiller]

It seems to me, to use the FDM in practice, you are going to have to be able to map a smoothly varying grid analytically onto the region of interest (using coordinate transformations, or whatever). FDM is not going to be very useful for very unusually shaped region boundaries, which is why FEM was really developed.

 

[bigfooted]

@Chestermiller makes a good point. Use FDM for rectangular domains. If you want to work with more complicated domains you can switch to the finite element method. Some additional links:

http://hplgit.github.io/INF5620/doc/pub/H14/fem/html/main_fem.html

https://fenicsproject.org/

 

[zoltrix]

hello

what is the advantage ,if any, of FDM / FEA over Mathematica / Maple to solve partial differential equations ?