In this paper we investigate the numerical solution of Poisson equations on adapted structured grids generated by multiresolution analysis. Such an approach not only involves important savings in computational costs, but also allows us to conduct a mathematical description of the numerical approximations in the context of biorthogonal wavelet decomposition. In contrast to most adaptive meshing techniques in the literature that solve the corresponding system of discrete equations level-wise throughout the set of adapted grids, we introduce a new numerical procedure, mainly based on inter-level operations, to represent in a consistent way the elliptic operators discretized on the adapted grid. In this way the discrete problem can be solved at once over the entire computational domain strongly coupling inter-grid relations as a completely separate process, independent of the mesh generation or any other grid-related data structure or geometric consideration, while the multiresolution framework guarantees numerical approximations within an accuracy tolerance. To assess the validity of both the theoretical estimates as well as the numerical construction of the discrete operators we investigate them in the context of streamer discharge simulations. The theoretical bounds are thus evaluated in a simpler configuration with analytical solution that nevertheless mimics the spatial structure found in this kind of problems; whereas the complete and more complex model is considered to simulate double-headed streamers with photoionization mechanisms, for which we study the performance and capabilities of various direct and iterative linear solvers.