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  "body": "\n## Definition\n\nLet $H$ be a real [[Hilbert space|HM2HHE3J51]], and let $B \\colon H \\times H \\to \\R$ be a bounded [[bilinear form|0SQZMXB6GP]].\nLet $f$ be a bounded linear functional on $H$, an element of the [[continuous dual space|4A82S5FVRT]].\nConsider the problem of finding $u \\in H$ with $B(u, v) = f(v)$ for all $v \\in H$.\nLet $V_N$ be a finite-dimensional subspace of $H$.\nThe *Galerkin method* replaces this problem by the following one: find $u_N \\in V_N$ such that\n$$B(u_N, v) = f(v) \\quad \\text{for all } v \\in V_N.$$\nThe element $u_N$ is the *Galerkin approximation* of $u$ in $V_N$.\n\n## Remarks\n\nIf $\\varphi_1, \\ldots, \\varphi_N$ 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 $i = 1, \\ldots, N$.\nIf $B$ is [[coercive|86KRHS6X22]] with constant $\\alpha$ and bounded with constant $M$, the [[Lax–Milgram theorem|85KK79FCN5]] applied on $V_N$ gives a unique $u_N$.\nSubtracting the two problems gives Galerkin orthogonality: $B(u - u_N, v) = 0$ for all $v \\in V_N$.\nFrom it follows Céa's lemma: $\\norm{u - u_N} \\le \\frac{M}{\\alpha} \\inf_{v \\in V_N} \\norm{u - v}$.\n\nExample: 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 $\\varphi(x) = x(1 - x)$.\nThen $B(\\varphi, \\varphi) = \\int_0^1 (1 - 2x)^2 \\, dx = 1/3$ and $f(\\varphi) = 1/6$, so $u_1 = \\frac{1}{2} x(1 - x)$.\nThis is the exact solution, as Céa's lemma predicts, since the exact solution lies in $V_1$.\n\nNon-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.\nIt is called a Petrov–Galerkin method, and its analysis needs an inf-sup condition instead of coercivity.\n",
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