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Difference between FEM, Galerkin and Discontinuous Galerkin

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Old   October 14, 2017, 06:02
Default Difference between FEM, Galerkin and Discontinuous Galerkin
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I need to do some work on Discontinuous Galerkin methods.

I do not much previous experience with FEM in general, so I am trying to understand what is the difference between FEM, Galerkin methods and DG methods.

At the moment, I gained impression that:

FEM -> Galerkin method -> Discontinuous Galerkin method, i.e. Galerkin method is a type of FEM method, and DG is special type of Galerkin method.

But what is it that differs Galerkin from general FEM framework, and what differs DG from standard Galerkin?
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Old   October 14, 2017, 09:44
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I heve no practical experience on FEM, G or DG methods but, what I remember is:

- FEM or, better, weighted residual methods assume your solution is written as an expansion in terms of so called trial functions. Coefficients multiplying trial functions are obtained by inserting the solution template in your equation, multiplying it for test function and integrating over volume of your domain.

- different test/trial functions lead to, practically, any known numerical method. Galerkin methods are those with test functions equal to trial functions (those of the expansion of the solution).

- I'm just guessing here but, I thing DG mostly differs by G in the class of functions allowed as test/trial. Where G only use continuous one, probably DG allows functions which are discontinuous across elements.

As I said, I have no practice on the FEM in general, but I expect that in practical terms (i.e., implementation) the difference might be more substantial (in terms of practical operations that you need to do in one method in contrast to the other).
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Old   October 14, 2017, 16:37
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Quote:
Originally Posted by sbaffini View Post
I heve no practical experience on FEM, G or DG methods but, what I remember is:

- FEM or, better, weighted residual methods assume your solution is written as an expansion in terms of so called trial functions. Coefficients multiplying trial functions are obtained by inserting the solution template in your equation, multiplying it for test function and integrating over volume of your domain.

- different test/trial functions lead to, practically, any known numerical method. Galerkin methods are those with test functions equal to trial functions (those of the expansion of the solution).

- I'm just guessing here but, I thing DG mostly differs by G in the class of functions allowed as test/trial. Where G only use continuous one, probably DG allows functions which are discontinuous across elements.

As I said, I have no practice on the FEM in general, but I expect that in practical terms (i.e., implementation) the difference might be more substantial (in terms of practical operations that you need to do in one method in contrast to the other).
Thank you for the reply!

I gained the same impression - that Galerkin is the one where basis functions are the same as test functions. I do not know if there is any other difference?

And also, that Discontinuous allows for - as the name says - discontinuous reconstructions.

Hopefully someone can confirm that this is true (or that these are major differences).
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Old   October 14, 2017, 17:07
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Quote:
Originally Posted by Amarant View Post
I need to do some work on Discontinuous Galerkin methods.

I do not much previous experience with FEM in general, so I am trying to understand what is the difference between FEM, Galerkin methods and DG methods.

At the moment, I gained impression that:

FEM -> Galerkin method -> Discontinuous Galerkin method, i.e. Galerkin method is a type of FEM method, and DG is special type of Galerkin method.

But what is it that differs Galerkin from general FEM framework, and what differs DG from standard Galerkin?

Galerkin does not differ from FEM framework but is a FEM based on a proper minimization of the error. The general idea is to define an error function that is orthogonal to the chosen subspace. We define Bubnov-Galerkin or Petrov-Galerkin depending on the choice of the projection.
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Old   October 15, 2017, 02:39
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Originally Posted by FMDenaro View Post
Galerkin does not differ from FEM framework but is a FEM based on a proper minimization of the error. The general idea is to define an error function that is orthogonal to the chosen subspace. We define Bubnov-Galerkin or Petrov-Galerkin depending on the choice of the projection.
Yes, my mistake, I did not mean that it is difference, but that is belongs to class of FEM. I was just not sure what are its major characteristics.

I am not that familiar with the way you described it - does it have to do something with basis functions being the same as test functions...?
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