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Assume we have few terms contributing to element tensor and each requires different quadrature degree to be integrated exactly. Does it generally worth using different quadrature degrees for each term or should one use the single one with highest degree for all terms?

I would guess that for sufficiently large $N$ (number degrees of freedom) it does not matter as solver cost being $O(N^\alpha)$ with $\alpha>1$ beats assembler cost $O(N)$.

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Use the highest order you need for all terms. The major expense is evaluating the values and gradients of the shape functions at the quadrature points, which you do once and then reuse as often as is necessary. If you use multiple different quadrature formulas, you end up with more quadrature points at which you need to evaluate shape functions, for a larger overall expense.

There are exceptions to this rule (sometimes you do want to purposefully under-integrate certain terms) but I think it's still generally true.

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  • $\begingroup$ Could you also add some example when is under-integration useful? $\endgroup$ – Jan Blechta Jun 7 '13 at 12:00
  • $\begingroup$ One sometimes does it with strongly varying coefficients. Another case is if you need to assemble a mass matrix and would like it to be diagonal. The typical case appears to be when you have a nearly incompressible elastic medium and want to avoid locking. But I will admit that I don't know very much about under-integration. I've never done it myself. $\endgroup$ – Wolfgang Bangerth Jun 10 '13 at 15:23

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