Solving Variational Equation w/ FreeFem++

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In summary, the conversation discusses the use of a specific form of the variational equation in freefem++ and the search for a resource to study it further. Recommendations for textbooks on numerical methods are given.
  • #1
urric
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Hi, sorry if this isn't the correct forum.
I'm trying freefem++ to solve some some problems with finite elements method, but in the documentation the following form of the variational equation is used:

[itex]\displaystyle\int_{T_{h}} (\frac{{\partial u}}{{\partial x}}\frac{{\partial v}}{{\partial x}} + \frac{{\partial u}}{{\partial y}}\frac{{\partial v}}{{\partial y}} )dxdy =\displaystyle\int_{T_{h}} fvdxdy[/itex]

I've searched for that form, but I wasn't able to find it

Could you tell me about a book or any source where I can study that form of the variational equation?

Thanks
 
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  • #2
in advance.This form of the variational equation is quite common and can be found in many textbooks on numerical methods, such as Finite Element Methods by Hughes (Dover, 2000) or Finite Element Analysis by Zienkiewicz and Taylor (Wiley, 2000).
 

1. What is the purpose of solving variational equations with FreeFem++?

The purpose of solving variational equations with FreeFem++ is to numerically find the solution to a differential equation that minimizes a given functional. This is useful in many areas of science and engineering, such as fluid dynamics, quantum mechanics, and structural analysis.

2. What is the difference between a variational equation and a differential equation?

A variational equation is derived from a functional, which is a mathematical expression that maps a set of functions to a real number. A differential equation, on the other hand, describes the relationship between a function and its derivatives. In solving variational equations, we are looking for the function that minimizes the given functional, rather than solving for a specific function as in traditional differential equations.

3. How does FreeFem++ solve variational equations?

FreeFem++ uses the finite element method to solve variational equations. This involves dividing the domain into smaller subdomains and approximating the solution using piecewise polynomial functions defined on these subdomains. The method then minimizes the functional by solving a system of linear equations derived from the variational equation.

4. Can FreeFem++ handle complex variational equations?

Yes, FreeFem++ can handle complex variational equations as long as they can be written in a weak form. This means that the equation must be expressed as a functional that can be minimized, rather than a specific function to be solved for. FreeFem++ also has built-in support for complex numbers in its finite element solver.

5. Is there a limit to the complexity of variational equations that can be solved with FreeFem++?

FreeFem++ is a powerful and flexible software that can handle a wide range of variational equations. However, there may be limitations in terms of the size and complexity of the problem that can be solved due to computational resources and memory constraints. It is best to consult the official documentation and user forums for advice on handling more complex problems.

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