"Procision uses what is called 'the external approximation method'. In the Galerkin method (which is used in the finite element method) the discrete solution space (fem space) is a subspace of the solution space (which has infinite dimensions). That means that every function that we can build using linear combinations of our finite element shape functions belongs to the solution space. In the external approximation method this is not true. There are functions in the discrete solution space that do not belong to the infinite dimensional solution space. This happens because in Procision the domain is chopped in several parts and the approximation is built on each of these parts (independently of each other). Then, at the interface of the parts, the discrete solution may be discontinuous (and therefore does not belong to the solution space). But then you can penalise these discontinuities using, for example, Lagrange multipliers and get a solution that is 'almost continuous' across the interface between two parts. The problem now is that you have to solve a saddle point problem because of the introduction of the Lagrange multipliers (and therefore live with the Ladyzhenskaya-Babuska-Brezzi conditioning and stability issues).
If the domain is not properly divided (you may call it meshed) then the matching of the approximations at the interfaces is more difficult.
The underlying method used in Procision is very similar to the Hybrid finite element method used by, e.g., Jaroslav Jirousek from the École Polytechnique Fédérale de Lausanne, Switzerland. He calls them Trefftz elements since he uses Trefftz functions inside each element (which corresponds to a part of the domain in Procision's jargon).
I hope this helps.... "
(Search for "Trefftz Elements". )