Finite element method

Remarks

In a basis $\varphi_1, \ldots, \varphi_N$ of $V_h$, the discrete problem is the linear system $A c = b$ with $A_{ij} = a(\varphi_j, \varphi_i)$ and $b_i = \ell(\varphi_i)$, and $u_h = \sum_j c_j \varphi_j$. If $a$ is coercive, $a(v, v) \ge \alpha \norm{v}^2$, and $\abs{a(u, v)} \le M \norm{u} \norm{v}$, then $u_h$ exists, is unique, and Céa's lemma gives $\norm{u - u_h} \le (M / \alpha) \min_{v_h \in V_h} \norm{u - v_h}$. Example: $-u'' = f$ on $(0, 1)$ with $u(0) = u(1) = 0$, $V = H^1_0(0, 1)$, $a(u, v) = \int_0^1 u' v'$, nodes $x_i = ih$ with $h = 1/(N+1)$, and hat functions $\varphi_i$ with $\varphi_i(x_j) = \delta_{ij}$. Then $\int \varphi_i'^2 = 2/h$ and $\int \varphi_i' \varphi_{i \pm 1}' = -1/h$, so $A = h^{-1} \operatorname{tridiag}(-1, 2, -1)$. Non-example: the Galerkin method with $V_N$ spanned by $\sin(\pi x), \ldots, \sin(N \pi x)$ satisfies clause 3 but not clause 2. These functions are not piecewise polynomials on a mesh, so this is a spectral method.