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The error estimate described here is the truncation error (also sometimes called the Galerkin error for the finite element method). It does not take into account:
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The quadrature error made by using numerical methods to approximate the finite element integrals.
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The geometrical approximation error made by representing the actual geometry by a polynomial representation (which is a sort of integration error for elements adjacent to or on a curved boundary).
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The algebraic error obtained by terminating the solvers prematurely (or by using a sloppy tolerance).
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Due to the independent maximum norms used for the dual error and the residual within each mesh element, the error estimate is normally an upper bound. When the error is very localized (to only a few elements) — for example, when a field value in a point is used as the functional — the discrepancy between the actual error and the estimated error tends to be larger than for cases where the error is less localized. For cases when the ppr method can be used, a rule-of-thumb is that the error estimate is accurate within a factor five when the error is not so localized and one order of magnitude larger when the error is very localized. When the gradient-based dual error estimation is used, the discrepancy can be much larger. This difference in accuracy occurs because this estimate does not have the correct asymptotic behavior (the correct convergence rate when the mesh size is diminished). A warning is given when the gradient method is used for a dependent variable.
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