Discrepancy of an approximation

One of the characteristics of the quality of an approximate solution $\overline u$ of an operator equation $P(u)=0$ (e.g. a linear algebraic system, a differential equation). The discrepancy is defined as the quantity $P(\overline u)$ or a norm of this quantity, e.g., $\|P(\overline u)\|_2$. If the estimate

$$\|u_1-u_2\|_1\leq C\|P(u_1)-P(u_2)\|_2$$

holds, then the error of the solution may be estimated in terms of the discrepancy:

$$\|\overline u-u\|_1\leq C\|P(\overline u)\|_2.$$

If no such estimate is available, the discrepancy provides an indirect indication of the quality of the approximate solution.

References

[1]	I.S. Berezin, N.P. Zhidkov, "Computing methods" , Pergamon (1973) (Translated from Russian)
[2]	N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from Russian)