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I know that it is possible to determine the theoretical order of accuracy (B) of numerical solutions in FVM (for instance for a steady problem in which only the central differencing scheme (CDS) is used to interpolate, B = 2 since CDS is 2nd order accurate). I know that with FVM the order of accuracy is obtained from the Taylor series.

I interested to know whether there is a method to obtain the convergence rate for FEM solutions.

I am also particularly interested to find out if there is a way of determining the order of accuracy for DGFEM. I have read in many papers that the convergence rate is normally p+1 where p is degree of the approximating polynomial. However I am uncertain as why this is not effected by the interpolation scheme used for the flux terms.

Thank you.

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  • $\begingroup$ Depends on what you mean by simple. In all cases, you make use of the approximation properties of polynomials to obtain a result of the form "if you use (local) polynomials of degree at most $p$ and if(!) your exact solution is $p$ times (weakly) differentiable, then the error is of order $p$". It's a matter of taste whether Taylor expansion (for FDM) is an easier way than the Bramble-Hilbert lemma (for FEM) to derive such an estimate. (It's true that DGFEM is a bit more complicated than FEM, though, but so is FVM compared to FDM). $\endgroup$ – Christian Clason Dec 22 '15 at 23:40
  • $\begingroup$ Or to put it another way, it's unclear what you're asking here. Could you try narrowing things down to a concrete question? $\endgroup$ – Christian Clason Dec 22 '15 at 23:45
  • $\begingroup$ Thank you. Please forget about which one is simpler, I guess that's a relative thing. Should not have said that ... I've changes the question slightly now. $\endgroup$ – melody Dec 23 '15 at 12:58
  • $\begingroup$ It's still not clear what answer you expect besides "Yes, of course, look at any textbook on FEM (e.g., by Braess) or DGFEM (e.g., by Di Pietro and Ern)." And of course the flux terms are part of the DGFEM discretization, so whether you obtain the optimal order or not depends on whether they are chosen appropriately (or not). However, these are usually not derived via interpolation of an independent quantity but by suitable modifications of the bilinear form, into which the (locally) polynomial approximation to the solution is inserted. $\endgroup$ – Christian Clason Dec 23 '15 at 13:27
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This depends on the problem you are solving.

Usually the elliptic and hyperbolic cases are handled separately because the spectral properties of their operators are very different, and so you have papers/books which answer this question for those specific cases.

Generally speaking the expectation is that if you use p-th order polynomials, then DGFEM/FEM will give you between p and p+1 order convergence. The basic strategy for proving this is fairly well understood, but it often needs to be modified when applying it in new situations.

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    $\begingroup$ To add to Reid's comment, order $p+1$ convergence is typically expected for the $L^2$ norm. In general, if you want convergence in a Sobolev norm, you lose one order for each degree of regularity (i.e. $H^1$ convergence is usually $p$, etc...) $\endgroup$ – Jesse Chan Dec 22 '15 at 23:56
  • $\begingroup$ Thank you. I have read in various papers that the expected order is normally p+1 in DGFEM, but does the order of accuracy not also depend on the interpolation schemes used at the interfaces for the flux terms? For instance if an upwind scheme is used rather than a linear interpolation scheme? $\endgroup$ – melody Dec 23 '15 at 12:54
  • $\begingroup$ I am reading more on this .., but on the note Reid made that the convergence rate would be between p and p+1, is it possible to obtain a p-1 convergence rate. The reason I am asking is that I used a 2nd order polynomial in a DG method to solve a 1D steady convection-diffusion problem, and tried different schemes to interpolate the flux terms, and I obtained a p-1 convergence rate with the number of nodes for one of the schemes, and p convergence rate for the others but I don't think I've done anything wrong. $\endgroup$ – melody Dec 23 '15 at 14:13
  • $\begingroup$ I'm not sure what you mean by linear interpolation scheme - can you explain? Also, what are the diffusion coefficient and boundary conditions? $\endgroup$ – Jesse Chan Dec 23 '15 at 18:41

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