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  "body": "\n## Summary\n\nThe hp-FEM is a version of the [[finite element method|98GZ9DVHTA]] that improves the approximation in two ways at once.\nIt refines the mesh, which makes the element size $h$ smaller, and it raises the polynomial degree $p$ on the elements.\nSmall elements are used near singularities of the solution, and high degrees where the solution is smooth.\nTake an elliptic problem with piecewise analytic data on a polygon.\nBabuška and Guo showed that a well chosen hp-sequence then converges exponentially in the number of unknowns.\nSee the [[definition|1F7NV6AT2X]].\n\n## Explanations\n\n### L1 human\n\nA standard finite element method uses many small pieces, each with a simple polynomial such as a straight line.\nThe hp-FEM also lets each piece use a curved polynomial of high degree.\nWhere the solution is smooth, high degree pays off quickly; near a corner where the solution has a kink in its derivatives, small pieces work better.\nMixing the two well gives errors that drop much faster than with either one alone.\n",
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