Euler Lagrange equation - weak solutions?

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SUMMARY

The discussion centers on the application of the Euler-Lagrange equation in the context of weak solutions and piecewise smooth functions. It highlights the potential for enriching the Euler-Lagrange approach to accommodate weak derivatives, particularly in finite element approximations. The conversation suggests exploring variational methods as a means to formulate finite element methods effectively. While no specific modern references are provided, the need for advanced texts on finite element methods is emphasized.

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  • Understanding of the Euler-Lagrange equation
  • Familiarity with weak derivatives
  • Knowledge of finite element methods (FEM)
  • Basic concepts of variational methods
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  • Research variational methods in finite element analysis
  • Study weak solutions in the context of differential equations
  • Explore advanced textbooks on finite element methods
  • Learn about piecewise smooth functions and their applications in FEM
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Mathematicians, engineers, and researchers involved in computational mechanics, particularly those focusing on finite element analysis and variational methods.

muzialis
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Hello there,

I was wondering if anybody could indicate me a reference with regards to the following problem.

In general, the Euler - Lagrange equation can be used to find a necessary condition for a smooth function to be a minimizer.
Can the Euler - Lagrange approach be enriched to cover piecewise smooth solutions, with a weak derivative?

Any reference or hint would be so much appreciated.

Thanks

Muzialis
 
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This is one way to create finite element approximations where the approximating function is piecewise smooth over the region (line, surface, or volume) covered by each element.

Look for something on variational methods of formulating finite elements.

Note, there are other ways to create FE approximations, which may appear to be mathematically "simpler", and avoid the difficuilty that for some applications of FE it is hard to find a variational form to minimise, but an "advanced" text on the math of FE methods should cover variational methods.

I learned this stuff a long time ago, so I can't give you a personal recommendation for a good modern textbook or website - sorry about that.
 

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