An Analysis of the Robust Convergent Method for a Singularly Perturbed Linear System of Reaction–Diffusion Type Having Nonsmooth Data

In this paper, a coupled system of [Formula: see text] second-order singularly perturbed differential equations of reaction–diffusion type with discontinuous source term subject to Dirichlet boundary conditions is studied, where the diffusive term of each equation is being multiplied by the small perturbation parameters having different magnitudes and coupled through their reactive term. A discontinuity in the source term causes the appearance of interior layers on either side of the point of discontinuity in the continuous solution in addition to the boundary layer at the end points of the domain. Unlike the case of a single equation, the considered system does not obey the maximum principle. To construct a numerical method, a classical finite difference scheme is defined in conjunction with a piecewise-uniform Shishkin mesh and a graded Bakhvalov mesh. Based on Green’s function theory, it has been proved that the proposed numerical scheme leads to an almost second-order parameter-uniform convergence for the Shishkin mesh and second-order parameter-uniform convergence for the Bakhvalov mesh. Numerical experiments are presented to illustrate the theoretical findings.

Addendum to: Exploring Hidden Local Ordering in Microemulsions with a Weak Directive Second Order Parameter

In a recent publication, my group discussed a directive second order parameter that hypothetically could form micrometer large structures that influence the rheological behavior of a bicontinuous microemulsion. For this, the viscosities of two microemulsions with the non-ionic surfactants C10E4 and C8E3 were determined over the wide frequency and shear rate range. Contrarily to our previous publications there are no elevated viscosities towards slowest motions of the rheometer. Thus, no micrometer large structures form in microemulsions. However, we argue and confirm that there are compartments with the size of several correlation lengths. This finding supports the development of a directional order parameter in microemulsions.