*"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).
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If the domain is not properly divided (you may call it meshed) then
the matching of the approximations at the interfaces is more
difficult.
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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).
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I hope this helps.... "*

(*Search for "Trefftz Elements".
*)