On the design of numerical methods

Many terms and ideas used in numerical methods have their origin in analytical mathematics. Despite the well-known discrepancies between number spaces of computers and those of good old mathematics, the consequence of applying mathematical theorems to numerical methods and the importance of physical reasoning are often underestimated. The objective of this paper is to demonstrate that useful numerical methods may violate analytic dogmas or prejudices. Moreover, it is shown how a careful analysis of the numerical problems can lead to better numerical methods, and what future numerical codes might look like.