Can the global (accumulated) discretization error be less than the local error during a step? Explain.

Local discretization error is the error that occurs during a single step of numerical integration, which arises because the numerical method is using an approximate solution instead of the true solution. The local error is a measure of the difference between the true solution and the approximate solution for a particular step. Global discretization error, on the other hand, is the accumulated error over all steps in the numerical integration process. It gives an overall idea of how accurate the numerical solution is compared to the true solution over the entire interval.

In numerical integration methods, the global error is an accumulation of local errors that occur during each step of the process. Since the local error is the error for a single step and the global error is the sum of these local errors for all steps, the global error is generally larger than a single local error. However, it may be possible that some of the local errors will cancel each other out (e.g., positive and negative errors) when estimating the global error. The method's error analysis would determine the exact impact of the local errors on the global error. In such cases, it is theoretically possible for the global error to be smaller than the local error for a particular step of the process. However, this situation is relatively unlikely.

In summary, while it is theoretically possible for the global (accumulated) discretization error to be smaller than the local error during a specific step, this situation is relatively rare. The global error is typically larger than the local error, as it accumulates the local errors across all steps in the numerical integration process. To have a smaller global error in such cases, local errors would need to cancel each other out in a significant manner, which is not frequently observed.