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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?
First of all, the problem you pose looks like the Ritz form of a finite element problem, but it is not, and it is weirding me out a bit. Furthermore, I think this is a homework question, your first few steps are wrong, and you are stuck because of that, so you are asking us to fix it. Giving you the benefit of the doubt, here is my answer.
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.
Do these and update your question above if you have further questions.
