Galerkin $L^2$ Projections

Question

I'm struggling with a question:

Consider the domain $Ω = (0, 1)$, and the space $V ⊆ L^2(Ω)$ spanned by the basis functions $v_1 (x) = 1$ $v_2 (x) = 1 + x$ Find the $Ψ^δ ∈ V$ which is the Galerkin $L^2$ projection of the function $Φ ∈ L^2(Ω)$ onto V , with $Φ (x) = x^2$

Any help would be appreciated

Answer 1

We know that $V=\langle\{v_1,v_2\}\rangle$, $\varphi=x^2$.

$\psi^\delta\in V$, so, there existe $\alpha_1,\alpha_2\in\mathbb{R}$ such that

$$\psi^\delta=\alpha_1 v_1+\alpha_2 v_2.$$

Since $\psi^\delta$ is the projection of $\varphi$ on $V$, it holds

$$(\varphi-\psi^\delta,v)=0\quad\textrm{ for all }v\in V,$$

or equivalently

$$(\varphi-(\alpha_1v_1+\alpha_2v_2),v_i)=0\quad\textrm{ for }v_i\in \{v_1,v_2\}$$

Now you have two equations with two unknowns. You can get $\alpha_1$ and $\alpha_2$ and with that the projection asked.