Towards multiscale functions: enriching finite element spaces with local but not bubble-like functions

In this paper we propose a novel way, via finite elements to treat problems that can be singular perturbed, a reaction–diffusion equation in our case. We enrich the usual piecewise linear or bilinear finite element trial spaces with local solutions of the original problem, as in the residual free bubble (RFB) setting, but do not require these functions to vanish on each element edge, a departure from the RFB paradigm. Such multiscale functions have an analytic expression, for triangles and rectangles. Bubbles are the choice for the test functions allowing static condensation, thus our method is of Petrov–Galerkin type. We perform several numerical validations which confirm the good performance of the method.