# Finite Element Method - 3D Mesh Generator - Metfem3D

## System of Nonlinear Equations

### Newton's method

The above written set of equations is of a nonlinear type. Let us denote all of them as

F(n1, …, nM) = 0

Thus to solve it we have to employ the iterative Newton's method[1]. It means that we have to start from an initial guess of {ni0}Mi=1 values.
And during next iterations for k = 0, 1, …

 nk+1 = nk - { F'(n1k, …, nMk) } -1 F(n1k, …, nMk)

the solution should be achieved. F'(n1k, …, nMk) denotes the following matrix of partial derivatives:

F'(n1k, …, nMk) = [
 ∂F1 / ∂n1k … ∂F1 / ∂nMk … … … ∂FM / ∂n1k … ∂FM / ∂nMk
]

where

∂Fb/∂nak = ki|zi|e / (ε0ε) {∑e K*b,ec}-1e Kb,ec ncke Kb,eac + D ∑e K*b,ea
+ 1/(Δt) ∑e Kb,ea + 2ki|zi|e / (ε0ε) {∑e K*b,ec}-1e Kb,ec nake Kb,eac

where c ≠ a.

### References

1. ^ C. T. Kelley, Solving Nonlinear Equations with Newton's Method in Fundamentals of Algorithms SIAM, 2003; J. Brzozka, L. Dorobczyński, MATLAB. Srodowisko obliczeń naukowo-technicznych. PWN, Warszawa, 2008