  This work is concerned with the quantification of the epistemic uncertainties
induced the discretization of partial differential equations. Following the
paradigm of probabilistic numerics, we quantify this uncertainty
probabilistically. Namely, we develop a probabilistic solver suitable for
linear partial differential equations (PDE) with mixed (Dirichlet and Neumann)
boundary conditions defined on arbitrary geometries. The idea is to assign a
probability measure on the space of solutions of the PDE and then condition
this measure by enforcing that the PDE and the boundary conditions are
satisfied at a finite set of spatial locations. The resulting posterior
probability measure quantifies our state of knowledge about the solution of the
problem given this finite discretization.
