proper weak form


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10 months ago by
Marcin  
This is a general question about FEM. Suppose I have a PDE of the from:
 $\nabla^2u+\left(a\cdot\nabla\right)u=0$2u+(a·)u=0 
Should the $\left(a\cdot\nabla\right)u$(a·)u  term be integrated by parts to obtain the weak form?
It would be great if someone could point me to a reference.
Thank you,
Marcin
Community: FEniCS Project

1 Answer


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10 months ago by
pf4d  
If your basis function -- called interpolation function, or shape function -- is differentiable up to the order of your PDE, you do not have to integrate by parts at all.  Integration by parts allows you to specify your weak form with a reduced-differentiable space.  So in this case, if you were to integrate your 2nd-derivative term by parts, you only require C^1 continuous basis to solve your BVP.  However, by increasing the order of the basis you increase the rate of convergence of your approximation to the true solution.  For a good introduction to finite elements, you might try Johnson, 1987:

http://store.doverpublications.com/048646900x.html
Thank you for a quick reply.
written 10 months ago by Marcin  
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