A priori bounds are derived for the discrete solution of second-order elliptic partial differential equations (PDEs). The bounds have two contributions. First, the influence of boundary conditions is taken into account through a discrete maximum principle. Second, the contribution of the source field is evaluated in a fashion similar to that used in the treatment of the continuous a priori operators. Closed form expressions are, in particular, obtained for the case of a conservative, second-order finite difference approximation of the diffusion equation with variable scalar diffusivity. The bounds are then incorporated into a resilient domain decomposition framework, in order to verify the admissibility of local PDE solutions. The computations demonstrate that the bounds are able to detect most system faults, and thus considerably enhance the resilience and the overall performance of the solver.