The numerical problem

To define the numerical problem properly we should be more precise about the requirements for the numerical approximation.

• Because the interesting features of the solution are the precise location and the precise shape of the different layers, we consider the $C(\Omega)$-norm the proper measure for the error.
• Not only the approximation should be given, but also an educated guess about the accuracy obtained.
• Because for small $\varepsilon$ the precise $C(\Omega)$-error is not easily computed, it is the responsibility of the scientist to give sufficient support for his claim about the accuracy obtained.

Practical considerations

• The idea is that a method is developed that can find a solution both for large and for arbitrarily small values of $\varepsilon$. For practical reasons we can restrict this range to $1\ge \varepsilon \ge 10^{-8}$.
  To compare different solution methods, we might restrict ourselves to a finite sequence of $\varepsilon$-values, viz., $\varepsilon =1, ~0.1,~ 0.05, ~10^{-2}, 10^{-3}, 10^{-4}, 10^{-6}$.
• The aim is to compute approximations to the solution in such a way that the error in the overall $C(\Omega)$-norm is smaller than a prescribed tolerance $\tau$ and the computational work (on a desktop computer) is minimized. Possible interesting tolerances, $\tau$, for the error in the $C(\Omega)$-norm might be (approximately) $\tau= 10^{-1}, ~10^{-2}, 10^{-3}, 10^{-4}, 10^{-6}$
• The solution should be computed within a reasonable time on a modern desktop computer. Minutes or in some cases hours may be considered reasonable times, days are not.

The choice of the method is completely free. Not only methods based on discretization are allowed, but also any other type, such as methods based on the analytic solution and asymptotic expansions thereof, provided the above mentioned criteria are satisfied.

Requirements on submissions for the BAIL 2008 competition

• The preferred representation of numerical solutions is described in the next section. If possible, the graphs along cuts should be submitted also in the form of ASCII files containing both the coordinates of the points used on the respective cut and the values of the solution at these points.
• For each computation an indication should be given of the computational resources used. For discretisation methods this should include the number of degrees of freedom involved.
• The method used has to be well described and relevant references should be given.