Numerical multiscale methods for a reaction-dominated model

A Galerkin enriched finite element method (GEM) is proposed for the singularly perturbed reaction–diffusion equation. This new method is an improvement on the Petrov–Galerkin enriched method (PGEM), where now the standard piecewise (bi)linear test space incorporates fine scales. This appears as the fundamental ingredient for suppressing oscillations in the numerical solutions. Also, new parameter-free stabilized finite element methods derived from both the GEM and the PGEM are driven by local generalized eigenvalue problems. In the process, jump stabilizing terms belonging to the class of CIP methods emerge as a result of the enriching procedure. Interestingly, numerical results indicate that jump-based stabilizations are unnecessary and sometimes undesirable when treating reaction-dominated problems. Finally, we establish relationships with more standard enriched and stabilized methods and show that the proposed methods outperform them numerically.