Numerical Analysis on Non-Smooth Problems: Some Examples

In a partial differential equation roughness in the solution may be caused in various ways, such as by i)a rough right hand side,ii)a rough boundary,iii)rough initial data. We shall briefly describe two examples from numerical analysis showing that:Example A) In a linear Poisson problem, roughness introduced by i) or ii) leads to dramatically different behavior in numerical solutions.Example B) Although for linear and semilinear parabolic problems, in the case of iii) (all other “data” being smooth) there is “no difference” in the sense that the solution is smooth for (some) positive time, in numerical analysis the linear and semilinear problem are miles apart.While the main point of Example A was “computational folklore” for many years before it was fully proven in 1984, the punchline of Example B was totally unexpected at the time of its discovery in 1987.More details, and more examples of rough stuff in the numerical analysis of partial differential equations, e.g., in singularly perturbed problems, can be found in WAHLBIN [7].