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Using finite volume method, Consider the problem: $-U_{xx}=f(x)$ for all $x$ in $(0,1)$ and $u(0)=u(1)=0$.
And consider the admissible mesh. Let $U^*i$ denotes the mean value of $U$ on $K_i($control volume$)$,

How can I show that $U^*i-U(x_i)=O(h^2$) where $h=\max(h_i)$?
Any help please

Math1995
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1 Answers1

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It generally depends on your integration scheme. The steps for calculating the error due to approximation are generally as follows:

  1. Determine the method you want to use. Finite volume has several ways in which it may integrate a problem, choose one of the methods. I presume you have that covered.

  2. Use a Taylor series approximation to determine the representation of the true value of $U(x_i)$ w.r.t $h_i$. I think in your case a forward and backward Taylor expansion will be required.

  3. Next, compare the integration scheme chosen (finite volume method) with Taylor expansion found (or a combination of the forward and backward Taylor expansion).

  4. Take the limit when $h_i$ becomes much smaller than one and obtain the order of magnitude.

For more information, it may be helpful to look up local truncation error (LTE).

Snifkes
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