### proper weak form

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This is a general question about FEM. Suppose I have a PDE of the from:

$\nabla^2u+\left(a\cdot\nabla\right)u=0$∇

Should the $\left(a\cdot\nabla\right)u$(

It would be great if someone could point me to a reference.

Thank you,

Marcin

$\nabla^2u+\left(a\cdot\nabla\right)u=0$∇

^{2}`u`+(`a`·∇)`u`=0Should 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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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

http://store.doverpublications.com/048646900x.html

Thank you for a quick reply.

written
10 months ago by
Marcin

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