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accuracy of weak solutions

Posted Sep 25, 2021, 11:53 a.m. EDT 3 Replies

Carl Xu
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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 comsol, the solution differs a lot from the analytic one locally while it's only accurate in its integral form (globally)?

3 Replies Last Post Sep 29, 2021, 5:00 p.m. EDT
Jeff Hiller COMSOL Employee

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Posted: 4 years ago Sep 27, 2021, 1:30 p.m. EDT

Hello Carl,

The finite element method is one of several numerical techniques used by COMSOL in arriving at an approximate solution to equations. You are correct that if employed improperly, any numerical method can yield an approximate solution that is further away from the exact solution than the user may presume. Theorists and practitioners have long studied how to quantify and minimize the error. At a minimum, performing a mesh refinement study is recommended. You will find many works on this topic in any serious textbook on the finite element method. There is a short introduction to the topic in the COMSOL Multiphysics Cyclopedia.

Best regards,

Jeff

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Jeff Hiller
Hello Carl, The finite element method is one of several numerical techniques used by COMSOL in arriving at an approximate solution to equations. You are correct that if employed improperly, any numerical method can yield an approximate solution that is further away from the exact solution than the user may presume. Theorists and practitioners have long studied how to quantify and minimize the error. At a minimum, performing a mesh refinement study is recommended. You will find many works on this topic in any serious textbook on the finite element method. There is a short introduction to the topic in the [COMSOL Multiphysics Cyclopedia](https://www.comsol.com/multiphysics/finite-element-method?parent=physics-pdes-numerical-042-62). Best regards, Jeff
Carl Xu
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Posted: 4 years ago Sep 28, 2021, 10:54 p.m. EDT

Hi Jeff, thanks so much for the info but what if mesh refinement doesn't help?

Hi Jeff, thanks so much for the info but what if mesh refinement doesn't help?
Jeff Hiller COMSOL Employee

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Posted: 4 years ago Sep 29, 2021, 5:00 p.m. EDT
Updated: 4 years ago Sep 29, 2021, 5:04 p.m. EDT

Hi Carl,

I tried to hint in my original message that inappropriate meshing is only one of various possible causes for the numerical (hence approximate) solution returned by an FEA solver to deviate from the exact (but usually unknown) solution of a PDE. Solver settings could be another (think for instance of what happens if you set convergence tolerances too loosely: you're then basically asking the software to content itself with a relatively inaccurate solution). And of course there are other possible causes, some of which are specific to particular physics, such as the presence of singularities in the analytical solution, etc, etc. There are no shortages of textbooks on the FEA method; Zienkiewicz, Hughes, Bathe, to name just a few of the classics.

Best,

Jeff

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Jeff Hiller
Hi Carl, I tried to hint in my original message that inappropriate meshing is only one of various possible causes for the numerical (hence approximate) solution returned by an FEA solver to deviate from the exact (but usually unknown) solution of a PDE. Solver settings could be another (think for instance of what happens if you set convergence tolerances too loosely: you're then basically asking the software to content itself with a relatively inaccurate solution). And of course there are other possible causes, some of which are specific to particular physics, such as the presence of [singularities](https://www.comsol.com/blogs/singularities-in-finite-element-models-dealing-with-red-spots/) in the analytical solution, etc, etc. There are no shortages of textbooks on the FEA method; Zienkiewicz, Hughes, Bathe, to name just a few of the classics. Best, Jeff

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