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  "body": "\n## Definition\n\nLet $V$ and $Q$ be real [[Hilbert spaces|HM2HHE3J51]] of functions on a bounded polygonal domain $\\Omega \\subset \\R^d$.\nLet $a \\colon V \\times V \\to \\R$ and $b \\colon V \\times Q \\to \\R$ be bounded [[bilinear forms|0SQZMXB6GP]], and let $f \\in V'$ and $g \\in Q'$.\nThe *mixed formulation* is the problem: find $(u, p) \\in V \\times Q$ with\n$$a(u, v) + b(v, p) = f(v) \\ \\text{ for all } v \\in V, \\qquad b(u, q) = g(q) \\ \\text{ for all } q \\in Q.$$\nA *mixed finite element method* for it consists of:\n\n1. a mesh $\\mathcal{T}_h$ of $\\Omega$;\n2. two [[finite element|98GZ9DVHTA]] spaces $V_h \\subseteq V$ and $Q_h \\subseteq Q$ of piecewise polynomial functions on $\\mathcal{T}_h$, chosen separately;\n3. the [[Galerkin|Y5B0PEBKE8]] problem: find $(u_h, p_h) \\in V_h \\times Q_h$ with the two equations above for all $v \\in V_h$ and $q \\in Q_h$.\n\n## Remarks\n\nExample: the mixed Poisson problem.\nFor $-\\Delta u = s$ with $s \\in L^2(\\Omega)$ in $\\Omega$ with $u = 0$ on $\\partial \\Omega$, set $\\sigma = \\nabla u$.\nTake $V = H(\\operatorname{div}, \\Omega)$, the vector fields in $L^2$ whose [[divergence|0MWV6ZTK6S]] is in $L^2$, and $Q = L^2(\\Omega)$.\nMultiplying $\\sigma = \\nabla u$ by $\\tau \\in V$ and integrating by parts with $u = 0$ on the boundary gives $\\int_\\Omega \\sigma \\cdot \\tau + \\int_\\Omega u \\, \\nabla \\cdot \\tau = 0$, and the equation gives $\\int_\\Omega (\\nabla \\cdot \\sigma) q = -\\int_\\Omega s q$.\nThis has the form above with $a(\\sigma, \\tau) = \\int_\\Omega \\sigma \\cdot \\tau$, $b(\\tau, q) = \\int_\\Omega q \\, \\nabla \\cdot \\tau$, $f = 0$ and $g(q) = -\\int_\\Omega s q$.\nA stable choice is the lowest-order Raviart–Thomas space for $V_h$ with piecewise constants for $Q_h$.\n\nNon-example: the standard method seeks $u_h$ alone, in continuous piecewise linear functions in the [[Sobolev space|M4CEZ5K9HP]] $H^1_0(\\Omega)$.\nIt has one unknown field and no form $b$, so it is not a mixed method.\n\nThe matrix of the discrete problem has the block form $\\begin{pmatrix} A \u0026 B^T \\\\ B \u0026 0 \\end{pmatrix}$, which is symmetric when $a$ is but never positive definite.\n",
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