A local projection type stabilization with exponential enrichments applied to one-dimensional advection–diffusion equations

We consider the local projection stabilization (LPS) for solving a singularly perturbed advection–diffusion two-point boundary value problem. In its classical one-level variant, the LPS uses polynomial bubble functions to enrich the standard finite element spaces of continuous, piecewise polynomial functions. As recently shown, the two-level approach can be considered also as a one-level method, however, with piecewise polynomial enrichments. Here, we study the question under which condition a linearly independent H1 function can serve as an enrichment for the standard space of continuous, piecewise polynomials of degree r leading to the same type of error estimates for the solution as the original one- and two-level approaches. Moreover, in the constant coefficient case, we derive formulas for the user-chosen stabilization parameter which guarantee that the piecewise linear part of the solution becomes nodal exact. Finally, we choose exponential enrichments based on the asymptotic expansion of the solution and show by numerical tests that compared to the classical one-level variant of the LPS – a considerable improvement of the accuracy of the solution on non-layer adapted meshes can be achieved.