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Difference between FEM, Galerkin and Discontinuous Galerkin 

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October 14, 2017, 06:02 
Difference between FEM, Galerkin and Discontinuous Galerkin

#1 
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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? 

October 14, 2017, 09:44 

#2 
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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). 

October 14, 2017, 16:37 

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Quote:
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). 

October 14, 2017, 17:07 

#4  
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Filippo Maria Denaro
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Quote:
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 BubnovGalerkin or PetrovGalerkin depending on the choice of the projection. 

October 15, 2017, 02:39 

#5  
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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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