  In a recent paper [{\em F. Bernal, J. Mor\'on-Vidal and J.A. Acebr\'on,
Comp.$\&$ Math. App. 146:294-308 (2023)}] an hybrid supercomputing algorithm
for elliptic equations has been put forward. The idea is that the interfacial
nodal solutions solve a linear system, whose coefficients are expectations of
functionals of stochastic differential equations confined within patches of
about subdomain size. Compared to standard substructuring techniques such as
the Schur complement method for the skeleton, the hybrid approach renders an
explicit and sparse shrunken matrix -- hence suitable for being substructured
again. The ultimate goal is to push strong scalability beyond the state of the
art, by leveraging the scope for parallelisation of stochastic calculus. Here,
we present a major revamping of that framework, based on the insight of
embedding the domain in a cover of overlapping circles (in two dimensions).
This allows for efficient Fourier interpolation along the interfaces (now
circumferences) and -- crucially -- for the evaluation of most of the
interfacial system entries as the solution of small boundary value problems on
a circle. This is both extremely efficient (as they can be solved in parallel
and by the pseudospectral method) and free of Monte Carlo error. Stochastic
numerics are only needed on the relatively few circles intersecting the domain
boundary. In sum, the new formulation is significantly faster, simpler and more
accurate, while retaining all of the advantageous properties of PDDSparse.
Numerical experiments are included for the purpose of illustration.
