  We develop a numerical strategy to solve multi-dimensional Poisson equations
on dynamically adapted grids for evolutionary problems disclosing propagating
fronts. The method is an extension of the multiresolution finite volume scheme
used to solve hyperbolic and parabolic time-dependent PDEs. Such an approach
guarantees a numerical solution of the Poisson equation within a user-defined
accuracy tolerance. Most adaptive meshing approaches in the literature solve
elliptic PDEs level-wise and hence at uniform resolution throughout the set of
adapted grids. Here we introduce a numerical procedure to represent the
elliptic operators on the adapted grid, strongly coupling inter grid relations
that guarantee the conservation and accuracy properties of multiresolution
finite volume schemes. The discrete Poisson equation is solved at once over the
entire computational domain as a completely separate process. The accuracy and
numerical performance of the method is assessed in the context of streamer
discharge simulations.
