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In finite element textbooks there are error estimate inequalities like $e<~h^p$ or $e<~N^{-p/d}$, where h is the FE length scale, d the # of dimensions, p the FE order, N the # of DOFs. Are these only valid for linear PDEs? If not, are these valid for all well defined PDEs with good regularity?

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What the exact form of these error estimates is depends on a number of factors:

  • What norm you measure the error in. When you state $e < h^p$, I assume you measure the error $e$ as the $H^1$ seminorm of the error $u-u_h$.
  • The regularity of the solution. If the solution is not sufficiently regular, then you cannot expect the full convergence rate.
  • The equation you consider. The estimate may be true for the Laplace equation, but not for first-order equations such as the advection equation. For nonlinear equations, you also often run into technical difficulties that lead to reduced convergence rates and/or the fact that our mathematical techniques are not good enough to actually prove anything.
  • The discretization you use.
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