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  "body": "\n## Definition\n\nLet $\\Omega \\subset \\R^d$ be a bounded polygonal domain, and let $V$ be a [[Sobolev space|M4CEZ5K9HP]] on $\\Omega$ such as $H^1_0(\\Omega)$.\nAn *hp finite element space* is given by two data:\n\n1. a mesh $\\mathcal{T}$ of $\\Omega$ into elements $K$;\n2. a degree vector $p = (p_K)_{K \\in \\mathcal{T}}$ of positive integers.\n\nThe space $V_{\\mathcal{T}, p}$ is the set of $v \\in V$ whose restriction to each $K$ is a polynomial of degree at most $p_K$.\nThe *hp-FEM* is the [[Galerkin method|Y5B0PEBKE8]] on a sequence of such spaces.\nIn this sequence, both the mesh and the degree vector may change from one step to the next.\n\n## Remarks\n\nThe $h$-version is the case of constant $p$ and finer meshes, and the $p$-version is the case of a fixed mesh and growing degrees.\nBoth are special cases of the [[finite element method|98GZ9DVHTA]] in this setting.\n\nExample: on $\\Omega = (0, 1)$ with a solution behaving like $x^\\lambda$ near $0$, with $0 \u003c \\lambda \u003c 1$, take a geometric mesh with nodes $0, \\sigma^{n-1}, \\ldots, \\sigma, 1$ for some $0 \u003c \\sigma \u003c 1$.\nTake degrees that grow linearly away from $0$.\nFor such geometric meshes, the energy error decays exponentially in a root of the number of unknowns $N$.\nIn two dimensions the rate is $\\exp(-b N^{1/3})$ for piecewise analytic data on polygons.\n\nNon-example: let $V = H^1_0(\\Omega)$ and take all functions that are polynomials of degree at most $p_K$ on each $K$, with no continuity across elements.\nA function with a jump across an interior face is not in $H^1(\\Omega)$, so this set fails the condition $v \\in V$.\nSuch spaces are used by hp discontinuous Galerkin methods instead.\n",
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