Pointwise nonlinear scaling for reaction–diffusion equations

Parabolic reaction–diffusion systems may develop sharp moving reaction fronts which pose a challenge even for adaptive finite element methods. We propose a method to transform the equation into an equivalent form that usually exhibits solutions which are easier to discretize, giving higher accuracy for a given number of degrees of freedom. The transformation is realized as an efficiently computable pointwise nonlinear scaling that is optimized for prototypical planar travelling wave solutions of the underlying reaction–diffusion equation. The gain in either performance or accuracy is demonstrated on different numerical examples.