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I have discretized the navier stokes equation in 2D but when I used Newton method to deal with the nonlinear, I don't know the shape of jacobian matrix (you can look at this finite element for incompressible Navier Stokes equation. p28-p30) function R is a vector with 15 column and have three part R1 R2 R3 so the Jacobian matrix must be a $15\times15$ matrix but how can you decide where is $\dfrac{\partial R1}{\partial u},\dfrac{\partial R1}{\partial v},\dfrac{\partial R1}{\partial p}$. I mean the dimension of each one will take up in Jacobian matrix

hung
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1 Answers1

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Have you tried AD? wikipedia. I do that for implicit method for gas dynamics with Newton and it work fine.

User1611161
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