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This will be a vague question.

When I was writing a finite element matrix assembly routine, a colleague noticed that I had a bug in my code because the sparsity pattern of the one of the blocks didn't look correct to him.

He explained to me that the block operator is a Laplacian operator, which is the discrete analogue of the continuous second derivative operator, hence there should be no coupling between the off diagonal block terms of the matrix and therefore that I had a bug in my code, since the sparsity pattern did not match. I was blown away by the alternative way of viewing the problem, and I want to learn more.

He didn't know of any reference materials that teach things like that; he just claimed to have learned it by experience. Does anyone know of a textbook or resource on discrete operators which can help train my brain to think about numerics in this sort of sense?

I come from a mainly computer science background, and have more coding intuition than this sort of mathematical intuition.

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    $\begingroup$ Put this in a different way: what was the reference you used to learn FEM? I would go as far as to say that FEM is precisely the rigorous study of discrete approximations to continuous operators. $\endgroup$ May 13, 2017 at 1:06
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    $\begingroup$ Although @RichardZhang has a point, I would add that if you make the exercise of discretizing several differential operators using finite elements and compare the "structure" of the resulting matrices you would increase the "intuition" that you are talking about. $\endgroup$
    – nicoguaro
    May 13, 2017 at 7:00
  • $\begingroup$ If you want a reference regarding your explicit question, you might try this. That's not an easygoing reading ... but it might help (maybe?). $\endgroup$
    – nicoguaro
    May 15, 2017 at 2:45

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