### (Deleted) Solving non-linear time-dependent system of PDEs with multiple boundary conditions (Nernst-Planck equation)

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7 weeks ago by
Dear the community members.

First of all, I apologize if there exists any of the duplicate questions already uploaded in the forum.
However, I kindly appreciate you to go over this question.
I am fairly new to the area of FEM and framework provided by Fenics project.

I am trying to solve Nernst-Planck equation and Poisson equation on the lane, but I wonder if this equation can be solved by using Fenics framework.

I would like to solve non-linear time-dependent system of PDEs (6 equations/variables), with both Dirichlet and Neumann boundary conditions applied.
The formula has a form of:
$d\frac{\partial u}{\partial t}-\nabla\cdot\left(c\nabla u\right)=f$dut ·(cu)=ƒ
where  $d=\left[1;1;1;1;1;0\right]$d=[1;1;1;1;1;0] ,  $c=\left[N\times0.39,N\times0.27,N\times1.02,N\times5.28,N\times0.16,1\right]$c=[N×0.39,N×0.27,N×1.02,N×5.28,N×0.16,1]  ( $N=10^{-4}\times60\times60\times24$N=104×60×60×24 ) and both are constants.

Except for  $f$ƒ   of the 6th element (which is 0),  $f$ƒ   of each element is calculated by:
$f_i=C_i\times c_i\times K_i\cdot\times\left(\frac{\partial u_i}{\partial x}+\frac{\partial u_i}{\partial y}\right)$ƒ i=Ci×ci×Ki·×(uix +uiy )  ( $\cdot\times$·×: elementwise multiplication)
where  $C=\left[1,1,-1,-1,2\right]$C=[1,1,1,1,2] which is a constant,  $K=\left[\frac{\left(\frac{\partial u_6}{\partial x}+\frac{\partial u_6}{\partial y}\right)}{F}+\sum_{i=1}^5\left(C_i\times c_i\times\left(\frac{\partial u_i}{\partial x}+\frac{\partial u_i}{\partial y}\right)\right)\right]\cdot\div\left[\sum_{i=1}^5\left(C_i^2\times c_i\times u_i\right)\right]$K=[(u6x +u6y )F +i=15(Ci×ci×(uix +uiy ))]·÷[i=15(Ci2×ci×ui)]  ( $\cdot\div$·÷: elementwise division) and  $F=9.648\times10^4$F=9.648×104 which is also a constant.

In addition, $u_5$u5  becomes  $u_5-\frac{\sum_{i=1}^4\left(u_i\times C_i\right)}{C_5}$u5i=14(ui×Ci)C5 .

I already managed to construct a mesh and define/apply the boundary conditions.

My questions are:
(1) Does Fenics project allow me to construct  $f$ƒ   containing  $u$u  and grad with multiplication and division? Could I consider this case as  $f$ƒ   is an ordinary constant?
(2) As I understood, we have to apply Green's function to all second derivatives of  $u$u , but  $f$ƒ seems too complex in this case. Is it okay to just go with the original equation (without Green's function)?

Any tips and advices will be greatly appreciated. Thank you very much in advance.
Community: FEniCS Project