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  "body": "\n## Definition\n\nLet $V$ be a real [[Hilbert space|HM2HHE3J51]] of functions on a bounded polygonal domain $\\Omega \\subset \\R^d$, for example the [[Sobolev space|M4CEZ5K9HP]] $H^1_0(\\Omega)$.\nLet $a \\colon V \\times V \\to \\R$ be a bounded [[bilinear form|0SQZMXB6GP]] and $\\ell \\colon V \\to \\R$ a bounded linear functional.\nThe [[weak form|SJZX81A783]] of the problem is: find $u \\in V$ with $a(u, v) = \\ell(v)$ for all $v \\in V$.\nThe *finite element method* consists of:\n\n1. A mesh $\\mathcal{T}_h$: a partition of $\\overline{\\Omega}$ into finitely many closed simplices of diameter at most $h$, meeting face to face.\n2. A finite-dimensional space $V_h \\subseteq V$ of functions that are polynomials of degree at most $k$ on each element of $\\mathcal{T}_h$, with a locally supported basis.\n3. The discrete problem: find $u_h \\in V_h$ with $a(u_h, v_h) = \\ell(v_h)$ for all $v_h \\in V_h$.\n\n## Remarks\n\nIn 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 $u_h = \\sum_j c_j \\varphi_j$.\nIf $a$ 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 $\\norm{u - u_h} \\le (M / \\alpha) \\min_{v_h \\in V_h} \\norm{u - v_h}$.\nExample: $-u'' = f$ 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 $\\varphi_i(x_j) = \\delta_{ij}$.\nThen $\\int \\varphi_i'^2 = 2/h$ and $\\int \\varphi_i' \\varphi_{i \\pm 1}' = -1/h$, so $A = h^{-1} \\operatorname{tridiag}(-1, 2, -1)$.\nNon-example: the Galerkin method with $V_N$ spanned by $\\sin(\\pi x), \\ldots, \\sin(N \\pi x)$ satisfies clause 3 but not clause 2.\nThese functions are not piecewise polynomials on a mesh, so this is a spectral method.\n",
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