Help with Understanding and Deriving NonlinearFormIntegrator::AssembleElementGrad

[j-signorelli]

Hello,

I have been working through understanding how to implement my own NonlinearFormIntegrator specifically for nonlinear scalar diffusion like $$\nabla \cdot (k(u) \nabla u) = 0$$

I have looked through #2791, #160, #2755, here, and the source code for DiffusionIntegrator and MassIntegrator but still have some confusions:

My main question is: How is, using the Gateaux derivative for determining the Jacobian, $\delta u$ actually represented in the code?

• For Nonlinear Poisson/Laplace solve #2791, the nonlinearity considered is a $f(u) = e^{\alpha u}$ term in a $(f(u), v)$ inner product. When I take the Gateaux derivative of $f(u)$, I get $F'(u, \delta u) = \alpha e^{\alpha u} \delta u$. However, the code uses $F'(u, \delta u) = \alpha e^{\alpha u} u$; why is this OK to just use $\delta u = u$?
• Similarly, for a diffusion operator $(\lambda \nabla u , \nabla v)$ represented by DiffusionIntegrator, my presumption is that the AssembleElementGrad function must implement $(F'(u, \delta u), \nabla v)$. Taking the Gateaux derivative of $f(u)=\lambda \nabla u$ gives me $F'(u, \delta u) = \lambda \nabla \delta u$. However, the code for DiffusionIntegrator::AssembleElementGrad simply just calls DiffusionIntegrator::AssembleElementMatrix, essentially assuming $\delta u = \phi_i$. Why is this acceptable?

Thank you for any assistance.

[vladotomov]

Hi @j-signorelli ,

We have $u = \sum_i u_i \phi_i$ for a FE function $u$. Then derivatives are taken w.r.t. the coefficients $u_i$:

$$\frac{\partial F(u)}{\partial u_i} = \frac{\partial F(u)}{\partial u} \frac{\partial u}{\partial u_i} = \frac{\partial F(u)}{\partial u} \phi_i$$

Does this help?

[j-signorelli]

Hi @vladotomov

Apologies for taking awhile to respond. Thank you - that is extremely helpful.

So if I understand correctly, for a nonlinear form $(e^{\alpha u}, v)$:
AssembleElementVector = $\int_\Omega e^{\alpha u} \phi_i d\vec{x}$
AssembleElementGrad = $\dfrac{\partial}{\partial u_j}\left(\int_\Omega e^{\alpha u} \phi_i d\vec{x}\right) = \int_\Omega \alpha e^{\alpha u}\phi_i \phi_j d\vec{x}$

Applying this for a DiffusionIntegrator, $(\lambda \nabla u, \nabla v)$, where $n$ sums over spatial dimensions:
AssembleElementVector = $\int_\Omega \lambda (\partial_n\phi_{i}) (\partial_n\phi_{j}) u_j d\vec{x}$
AssembleElementGrad = $\dfrac{\partial}{\partial u_k}\left(\int_\Omega \lambda (\partial_n\phi_i)(\partial_n\phi_j) u_j d\vec{x}\right) = \int_\Omega \lambda (\partial_n\phi_i)(\partial_n\phi_{j}) d\vec{x}$ = AssembleElementMatrix

Applying this for a nonlinear diffusion integrator, $(k(u) \nabla u, \nabla v)$:
AssembleElementVector = $\int_\Omega k(u) (\partial_n\phi_i) (\partial_n\phi_j) u_j d\vec{x}$
AssembleElementGrad = $\dfrac{\partial}{\partial u_k}\left(k(u) (\partial_n\phi_i) (\partial_n\phi_j) u_j d\vec{x}\right) = \int_\Omega k'(u) (\partial_n\phi_i)(\partial_n\phi_j) u_j \phi_k d\vec{x} + \int_\Omega k(u) (\partial_n\phi_i)(\partial_n\phi_k) d\vec{x}$

Is my understanding correct now for this?

Thanks again very much.

[vladotomov]

Is this in Einstein notation @j-signorelli?
If yes, then it looks correct.