I know a notion of smoothness for functions, say, in $\mathbb{R}^n$, which simply means of class $C^\infty$. But in studying Numerical Analysis I sometimes read the term smooth for discrete functions, so I want to know what's the correct meaning of it. In the case which is of my interest, I am studying multigrid methods, and I always read that after $k$ iterations of Jacobi / SOR, the error and the residual are smooth, but the error is the difference between the exact solution and the approximated one. The latter is discrete in the sense that its domain of definition is the set of mesh points of a grid. So, what does smooth mean in this context?

  • $\begingroup$ How is that error discrete? What is the smallest positive value it can attain, for example? $\endgroup$ Jul 7, 2018 at 11:01
  • $\begingroup$ "Smooth" means whatever the author of the text wants it to mean. This question cannot be objectively answered. $\endgroup$
    – user357151
    Jul 7, 2018 at 19:36


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