Control of Numerical Errors

True error (based on an exact solution) is not usually available – if we had this, why would we even use numerical methods? Thus, you generally want to use approximate (estimated) error instead like:

\begin{align}
\epsilon_{a}&=\frac{\text{approximate error}}{\text{approximate value}} \cdot 100\% \\ \\
\epsilon_{a}&=\frac{\text{current approximation}-\text{previous approximation}}{\text{current approximation}} \cdot 100\%
\end{align}

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No systematic, general approach for error estimation for all problems, as specific methods use different approaches.

Some guidelines:

• Avoid subtracting two nearly equal numbers
• Do not add very small and very large numbers together

Error control methods:

• Sensitivity analysis, such as grid refinement study or sensitivity to variations in input parameters
• Examine limiting cases (e.g. upper/lower bounds)