In this talk, we'll consider the numerical approximation of singularly perturbed reaction-diffusion partial differential equations, by finite element methods (FEMs).

Solutions to such problems feature boundary layers, the width of which depends on the magnitude of the perturbation parameter. For many hears, some numerical analysts have been preoccupied with constructing methods that can resolve any layers present, and for which one can establish an error estimate that is  independent of the perturbation parameter. Such methods are called "parameter robust", or (in some norms) "uniformly convergent".

In this talk we'll begin with the simplest possible parameter robust FEM: a standard Galerkin finite element method (FEM) applied on a suitably constructed  mesh using a priori information. However, from a practical point of view, not very scalable. To resolve this issue we consider the application of sparse grid techniques. These methods have many variants, two of which we'll consider: the hierarchical basis approach (e.g., Zenger, 1991) and the
two-scale method (e.g., many papers by Aihui Zhou and co-authors). The former can be more efficient, while the latter is considered simpler in both theory and practice.

Our goal is to try to unify these two approaches (at least in two dimensions), and then extend to three-dimensional problems, and, moreover, to other FEMs.