Galerkin method

Remarks

If $\varphi_1, \ldots, \varphi_N$ is a basis of $V_N$ and $u_N = \sum_j c_j \varphi_j$, the problem is the linear system $\sum_{j=1}^{N} B(\varphi_j, \varphi_i) \, c_j = f(\varphi_i)$ for $i = 1, \ldots, N$. If $B$ is coercive with constant $\alpha$ and bounded with constant $M$, the Lax–Milgram theorem applied on $V_N$ gives a unique $u_N$. Subtracting the two problems gives Galerkin orthogonality: $B(u - u_N, v) = 0$ for all $v \in V_N$. From it follows Céa's lemma: $\norm{u - u_N} \le \frac{M}{\alpha} \inf_{v \in V_N} \norm{u - v}$.

Example: for $-u'' = 1$ on $(0, 1)$ with $u(0) = u(1) = 0$, take $H = H^1_0(0, 1)$, $B(u, v) = \int_0^1 u' v' \, dx$, $f(v) = \int_0^1 v \, dx$ and $V_1$ spanned by $\varphi(x) = x(1 - x)$. Then $B(\varphi, \varphi) = \int_0^1 (1 - 2x)^2 \, dx = 1/3$ and $f(\varphi) = 1/6$, so $u_1 = \frac{1}{2} x(1 - x)$. This is the exact solution, as Céa's lemma predicts, since the exact solution lies in $V_1$.

Non-example: choosing $u_N \in V_N$ so that $B(u_N, w) = f(w)$ for all $w$ in a different subspace $W_N \ne V_N$ is not the Galerkin method in the sense above. It is called a Petrov–Galerkin method, and its analysis needs an inf-sup condition instead of coercivity.