Weak solutions under finite elements

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Discussion Overview

The discussion revolves around the nature of weak solutions in finite element methods (FEM) and their relationship to classical (strong) solutions of partial differential equations (PDEs). Participants explore the implications of numerical solutions provided by finite element software, particularly in terms of local discrepancies from classical solutions while maintaining global accuracy in integral form.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning
  • Experimental/applied

Main Points Raised

  • Some participants propose that weak solutions can differ significantly from classical solutions locally, even if they are accurate in an integral sense globally.
  • Others clarify that if a classical solution exists, it is also a weak solution, and if weak solutions are unique, they coincide with the classical solution.
  • A participant notes that weak solutions may admit discontinuities, which could lead to significant local differences from classical solutions.
  • Concerns are raised about numerical errors in software, including issues like hourglassing and shear locking, which may affect the accuracy of finite element analyses.
  • Some participants mention specific algorithms and element types, questioning their susceptibility to numerical issues.
  • References to literature on finite element procedures are provided as potential resources for understanding convergence and error problems.
  • It is noted that modeling inaccuracies, such as discretization and boundary conditions, contribute significantly to overall error in finite element analyses.

Areas of Agreement / Disagreement

Participants express varying views on the relationship between weak and classical solutions, with some agreeing on the definitions while others highlight the potential for discrepancies. The discussion remains unresolved regarding the extent of local differences in numerical solutions and the impact of specific numerical methods.

Contextual Notes

Limitations include assumptions about the accuracy of software in reproducing weak solutions, the definitions of classical versus weak solutions, and the potential for numerical errors due to modeling choices.

Who May Find This Useful

This discussion may be of interest to practitioners and researchers in finite element analysis, particularly those dealing with numerical methods, error analysis, and the theoretical underpinnings of weak versus classical solutions in PDEs.

feynman1
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Finite elements give weak solutions, that is, the solutions to a PDE are only correct in its integral form. Is it possible that in finite element software, the solution differs a lot from the analytic one locally while it's exact in its integral form (globally)?
 
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By "analytic" do you mean "classical", as in: satisfying the differential form of the PDE?

A classical solution need not exist, and weak solutions may admit discontinuities. In that case, a weak solution would be very far from "differentiable", and the answer to your question would be "yes".

On the other hand, if a classical solution exists, then it is also a weak solution. If, moreover, weak solutions are unique, then the classical solution and the weak solution are the same, and course the answer to your question would be "no".

(This all assumes that the software accurately reproduces the weak solution, including its possible discontinuities.)
 
S.G. Janssens said:
By "analytic" do you mean "classical", as in: satisfying the differential form of the PDE?

A classical solution need not exist, and weak solutions may admit discontinuities. In that case, a weak solution would be very far from "differentiable", and the answer to your question would be "yes".

On the other hand, if a classical solution exists, then it is also a weak solution. If, moreover, weak solutions are unique, then the classical solution and the weak solution are the same, and course the answer to your question would be "no".

(This all assumes that the software accurately reproduces the weak solution, including its possible discontinuities.)
Sorry there was a typo in my original post, should change 'analytic' to 'strong'.
Let's not consider discontinuities.
If a classical/strong solution exists, then why is it also a weak solution?
Yes, this post considers numerical errors by software. So it's asking whether software numerical solutions can give weak solutions differing a lot from the strong one locally while it's accurate in its integral form (globally).
 
There are things like hourglassing and shear locking.
 
caz said:
There are things like hourglassing and shear locking.
I use quadratic elements and U-P nearly impressible algorithm, will these still happen?
 
I’m not so much into FEM theory but you may find the answer in "Finite Element Procedures" by K.J. Bathe or "Concepts and Applications of Finite Element Analysis" by R.D. Cook. These two books cover convergence and error problems in detail.

In practice, I wouldn’t worry about that. There’s always an error (at least a few percent discrepancy from actual behavior of the structure) but it’s mainly caused by discretization, inaccurate material properties, boundary conditions that do not fully represent real-life supports and so on. Basically, largest part of this total error is caused by imprecise modeling.

When it comes to hourglassing and locking, let’s summarize them shortly:
- hourglassing - occurs when bending is analyzed using first order elements with reduced integration
- shear locking - occurs when bending is analyzed using first order elements with full integration
- volumetric locking - occurs when incompressible or nearly incompressible materials are analyzed, especially using elements with full integration
 
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