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

• I
• zoltrix
In summary, FDM and FEM are two different methods for solving differential equations. There are advantages and disadvantages to each method.
zoltrix
hello

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

Delta2
To avoid any possible confusion here, what do you mean by FDM and FEM?

I’m guessing PDE is for partial differential equations.

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

berkeman
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
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

zoltrix said:
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.

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

is FDM suitable for all these types of PDE ?

Delta2
Yes

Delta2
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 ?

Thanks bigfooted

it is exactly what I was looking for

berkeman
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
hello

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

## 1. Can FDM solve any type of PDE?

No, FDM (Finite Difference Method) is not able to solve all types of PDEs (Partial Differential Equations). It is best suited for solving linear, elliptic, and parabolic PDEs, but it may not be as accurate for solving hyperbolic PDEs.

## 2. How does FDM compare to FEM in terms of solving PDEs?

FDM and FEM (Finite Element Method) are both numerical methods used to solve PDEs. FDM approximates the derivatives of the PDE at discrete points in the domain, while FEM approximates the solution over a mesh of smaller elements. FDM is generally easier to implement, but FEM may provide more accurate solutions for complex PDEs.

## 3. Are there any limitations to using FDM for solving PDEs?

Yes, there are some limitations to using FDM. It may not be as accurate for solving PDEs with irregular or complex geometries, and it may require a large number of grid points to achieve a desired level of accuracy. Additionally, FDM may struggle with solving PDEs with discontinuous coefficients or boundary conditions.

## 4. Can FDM be used for time-dependent PDEs?

Yes, FDM can be used to solve time-dependent PDEs, such as parabolic PDEs. However, it may require additional techniques, such as the Crank-Nicolson method, to handle the time-dependent aspect of the PDE.

## 5. Is FDM the best method for solving PDEs?

There is no one "best" method for solving PDEs, as it depends on the specific problem and its characteristics. FDM is a popular and widely used method, but other methods such as FEM, spectral methods, and finite volume methods may be more suitable for certain types of PDEs or certain types of problems.

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