Derive the weak form for nonlinear problem.

Question

Let the equation be $$\frac{d^2 y}{dx^2}=\frac{y}{1+y}$$ For finite element formulation how to get the weak form? The major problem being the nonlinear rhs.`

Answer 1

Let me rewrite your expression as \begin{align} y_{xx} = \frac{y}{1+y}\Leftrightarrow y_{xx} +yy_{xx} -y = 0 \end{align} Since \begin{align} \Bigl(\frac{y^2}{2}\Bigr)_{xx} = (yy_x)_x = y_x^2+yy_{xx} \Leftrightarrow yy_{xx} = \Bigl(\frac{y^2}{2}\Bigr)_{xx}-y_x^2 \end{align} you can rewrite the first expression as \begin{align} y_{xx} +yy_{xx} -y = 0 \Leftrightarrow y_{xx} +\Bigl(\frac{y^2}{2}\Bigr)_{xx}-y_x^2-y = 0 \end{align} Assume, that $\phi^i$ are our (standard) testfunctions (which vanish on $\partial \Omega$). For the weak formulation we project onto the testspace. Let $\Omega$ be our domain, we then have for all $i$ \begin{align} &\int_{\Omega}\Bigl[ y_{xx} +\Bigl(\frac{y^2}{2}\Bigr)_{xx}-y_x^2-y \Bigr]\phi^i\,dx = 0\\ \Leftrightarrow& -\int_{\Omega}\Bigl[y_x+\frac{y^2}{2}\Bigr]\phi^i_x\, dx -\int_\Omega \Bigl[y^2_x+y \Bigr] \phi^i\,dx=0 \\ \Leftrightarrow & \int_{\Omega}\Bigl[y_x+\frac{y^2}{2}\Bigr]\phi^i_x\, dx +\int_\Omega \Bigl[y^2_x+y \Bigr] \phi^i\,dx=0 \end{align} Now you have only derivatives of order zero and one and can use $y\approx y^N$ as it is always done with FEM.