# Discretization of a nonlinear boundary value problem

I am trying to use finite element method to discretize the following problem \begin{align} \min_{u \in H^1_0(\Omega)} \int \| \Delta u(x) - 0.5*[u(x) + \langle e, x \rangle + 1]^3 \|^2_2 \ d\Omega, \end{align} where e is the vectors of ones. The solution $$u$$ is scalar-valued.

It is not clear to me, how should one handle squared norm. Should the whole term inside be multiplied with the test function?

• Please clarify: Is $u$ vector-valued? I suggest deriving the optimality condition (i.e., set the first variation to $0$). This yields a variational problem that can be discretized as usual. – cos_theta Mar 7 at 0:09
• thanks, I added the information. – enrico Mar 7 at 9:47
• If $u$ is a scalar quantity, what does the norm inside the integral mean? If this is $L_2$-norm, then what does the integral mean (domain of integration, variable of integration)? – cos_theta Mar 7 at 11:12
• integral goes over domain $\Omega$ – enrico Mar 7 at 11:21
• I still don't get it: $\Delta u(\vec{x}) - (1/2) [u(\vec{x}) + \langle \vec{e}, \vec{x} \rangle + 1]^3$ is a function $\Omega \to \mathbb{R}$ (since $u$ is scalar-valued). Taking the $L_2(\Omega)$-norm ($\lVert \Delta u(\vec{x}) - (1/2) [u(\vec{x}) + \langle \vec{e}, \vec{x} \rangle + 1]^3 \rVert_{L_2(\Omega)}^2 \in \mathbb{R}$) yields a constant. So the integral over $\Omega$ only multiplies that $L_2(\Omega)$-norm by $\mathrm{vol}(\Omega)$? – cos_theta Mar 7 at 13:44

Solving the problem \begin{align} \min_{u \in H^1_0(\Omega)} \int \| \Delta u(x) - 0.5*[u(x) + \langle e, x \rangle + 1]^3 \|^2_2 \ d\Omega, \end{align} is equivalent to finding $$u \in H^1_0(\Omega)$$ such that \begin{align} \Delta u(x) - 0.5*[u(x) + \langle e, x \rangle + 1]^3&=0 & \text{ in } \Omega, \\ u(x) &= 0 & \text{ on } \partial\Omega. \end{align} This should be straight-forward because if such $$u$$ exists then $$\int_{\Omega} \| \Delta u(x) - 0.5*[u(x) + \langle e, x \rangle + 1]^3 \|^2_2 \ d\Omega = 0$$ and obviously that gives you the minimum. From this point on, you can discretize the PDE as usual, but the PDE is non-linear due to the term $$0.5*[u(x) + \langle e, x \rangle + 1]^3$$. You should linearize it somehow. Picard iterations would be easy, but I suspect that they will not converge you need to do analysis. Newton's method would converge for a good initial guess, but you need to derive the method yourself.