  Under compressive creep, visco-plastic solids experiencing internal mass
transfer processes have been recently proposed to accommodate singular cnoidal
wave solutions, as material instabilities at the stationary wave limit. These
instabilities appear when the loading rate is significantly faster than the
capability of the material to diffuse internal perturbations and lead to
localized failure features (e.g., cracks and compaction bands). This type of
solution, generally found in fluids, has strong nonlinearities and periodic
patterns. Due to the singular nature of the solutions, the applicability of the
theory is currently limited. Additionally, effective numerical tools require
proper regularization to overcome the challenges that singularity induces. We
focus on the numerical treatment of the governing equation using a nonlinear
approach building on a recent adaptive stabilized finite element method. This
method provides a residual representation to drive adaptive mesh refinement, a
particularly useful feature for the problem at hand. We compare against
analytical and standard finite element solutions to demonstrate the performance
of our approach. We then investigate the sensitivity of the diffusivity ratio,
main parameter of the problem, and identify multiple possible solutions, with
multiple stress peaks. Finally, we show the evolution of the spacing between
peaks for all solutions as a function of that parameter.
