# finite element non linear boundary value problem

I have the following small finite element non linear boundary value problem:

$$-u''(x) = 1 + u^2(x) \quad \text{for} \quad 0

with a grid with points at $$\vec{x}= (0.2, 0.4, 0.6, 0.8)$$

The system of equation can be written as followed :

$$\frac{1}{0.2^2}\begin{bmatrix} +2 & -1 & 0 & 0 \\ -1 & +2 & -1 & 0 \\ 0 & -1 & +2 & -1 \\ 0 & 0 & -1 & +2 \\ \end{bmatrix} \begin{pmatrix} u_1 \\ u_2 \\ u_3 \\ u_4 \\ \end{pmatrix} = \begin{pmatrix} 1 + u_1^2 \\ 1 + u_2^2 \\ 1 + u_3^2 \\ 1 + u_4^2 \\ \end{pmatrix}$$ which can also be written as $$\textbf{A}\cdot\vec{u} = 1 + \vec{u}^2$$

Using the method of partial substitution to solve the problem with an initial vector $$\vec{u_0}$$ e.g. $$\vec{u_0}=\vec{0}$$. We can then use the iteration

$$\textbf{A}\cdot\vec{u}_{k+1} = 1 + \vec{u}_k^2$$ $$\vec{u}_{k+1} = \textbf{A}^{-1}\cdot\ (1 + \vec{u}_k^2)$$

my question is : why is there an increment of 1 in the factor $$k$$ in the last 2 equations? why is it not

$$\textbf{A}\cdot\vec{u}_{k} = 1 + \vec{u}_k^2$$

• How would you do the iterations if you had $u_k$ on both sides? – Yuriy S Nov 24 '18 at 19:27
• thank you for your comments @Yuriy S. Regarding your second comment you're write. I corrected the error – ecjb Nov 24 '18 at 19:31

You're making a sequence of progressively better solutions. $$\vec{u}_{k+1}$$ is the $$k+1^{\text{th}}$$ approximation and you have a recipe to construct it from the $$k^{\text{th}}$$ approximation.