# Can FDM solve any type of PDE same as FEM?

• I
zoltrix
hello

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

• Delta2

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

I’m guessing PDE is for partial differential equations.

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

• berkeman
Gold Member
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.

• • jasonRF and Delta2
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

Gold Member
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 ?

• Delta2
Gold Member
Yes

• Delta2
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 ?

Gold Member
• berkeman
zoltrix
Thanks bigfooted

it is exactly what I was looking for

• berkeman
Mentor
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
zoltrix
hello

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