Numerical methods that have basis functions (explicit or not) can be described and analysed either through the lens of these basis functions ("continuous" analysis), or via an approach solely based on the degrees of freedom ("fully discrete" analysis). In this presentation, I will start by developing these two approaches, and discuss their similarities and differences - in particular in terms of the underlying assumptions and possible challenges or limitations.
I will then consider the Discrete De Rham complex, for which basis functions can be identified, but are not computable and are not known to satisfy the required properties identified in the continuous analysis approach. I will discuss how the fully discrete approach was, however, successfully applied to this method, and illustrate its results on the curl-curl formulation of the Stokes equations.