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If we know the exact solution and approximate solution of a PDE then we can compute the error of both solutions in the discrete norm as given: $\|u-u_N\|^2_M$ = $\frac{2\pi}{M}\sum _{j=0}^{M-1}(u(z_j, t)-u_N(z_j, t))^2$. I am unable to understand this relation... kindly explain.

Reference: it is written in the book Implementing Spectral Methods for Partial Differential Equations by David A. Kopriva

Svetoslav
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R. Shah
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  • What is it you do not understand? – Friedrich Philipp Feb 27 '16 at 17:28
  • I am unable to understand the only equation mentioned above i.e $|u-u_N|^2_M$ = $\frac{2\pi}{M}\sum _{j=0}^{M-1}(u(z_j, t)-u_N(z_j, t))^2$ – R. Shah Feb 27 '16 at 17:33
  • Well, if you define $|f|M := \frac{2\pi}{M}\sum{j=0}^{M-1}|f(z_j,t)|^2$ for a function $f : \mathbb R^2\to\mathbb R$ then you get what's there, right? So, what's your point? The equation's interpretation? – Friedrich Philipp Feb 27 '16 at 17:45
  • in discrete fourier transform (DFT) why we set h =$ \frac { 2\pi}{M}$, $z_j = jh=\frac{{ 2\pi}j}{M}$ and takes summation $\sum _{j=-M/2}^ {j= M/2-1}$ . – R. Shah Feb 28 '16 at 02:11

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