How to generate functional in FEM

[mdn]

Please suggest me, how to generate weak form or functional of any partial diffrential equation ( mostely second order) in Finite Element Method.
Thanks in advance.

 

[AlephZero]

There are several methods, for example
1. Use something related to the physics of the model, for example a stationary value of energy.
2. Use calculus of variations.
3. Use an empirical method like the Galerkin method.

If several different methods give you the same formulation, that is usually good news. Otherwise, you may have to investigate which method is "best" in different situations.

 

[mdn]

thanks AlephZero to reply me, but actually i developed finite element solver (1D and 2D)in Java.
It can handle governing equation like: Poisson's , Laplace and Helmholtz equation.
I know how to generate weak form of above equations(from book reference), but don't have confident for other equations, that's why i would like to know, is there any general rule to make weak form, from strong form?
Again i confused about variational formulation, List square and Galerkin method.

 

[AlephZero]

is there any general rule to make weak form, from strong form?
Again i confused about variational formulation, List square and Galerkin method.

I think that question is too general to give an answer that is short enough for an internet forum. The answer could be a chapter of a textbook, or even a whole book.

Also we don't know what your math background is. A "cookbook" method for making a simple FE approximation to a specific equation would look very different from a general discussion in terms of functional analysis.

I think you need to ask more specific questions, to get useful answers.

 

[timthereaper]

Totally agree with AlephZero. I remember learning the finite element method and it taking months to do, and apparently we'd only scratched the surface. The biggest takeaway from learning FEM for me is that it's a method, not a recipe. It won't tell you exactly how to generate weak forms from strong forms or what approximations to make, but it can tell you what to do next when you have a weak form, for example.

 

[mdn]

Totally agree with AlephZero. I remember learning the finite element method and it taking months to do, and apparently we'd only scratched the surface. The biggest takeaway from learning FEM for me is that it's a method, not a recipe. It won't tell you exactly how to generate weak forms from strong forms or what approximations to make, but it can tell you what to do next when you have a weak form, for example.

but it is very simple now, just multiply by test function to strong form and integrate by parts in one dimension, this method will automatically make week form in one dimension.
and for two dimensional case, we can't integrate by parts in surface integration that's why we have to use stokes and divergence theorem. that's it.
this the thumb rule for any variational method.
any suggestion appreciated.