Lean Complementarity for Poisson Problems

We introduce a novel technique-lean complementarity-that attempts to eliminate any waste of computational resources occurring during the pursuing of complementarity. First, contrarily to the widely used practice of solving the problem two times with a pair of complementary or complementary-dual formulations, lean complementarity requires just one solution with the computationally cheap formulation based on the scalar potential. This result is enabled by a novel and explicit flux equilibration technique that produces tight bounds and is computationally inexpensive, because no system has to be solved. Second, the systems arising during the adaptive mesh refinement procedure are solved inexactly on purpose, by stopping the iterations of the iterative solver when the algebraic error gets negligible with respect to the discretization error. The discretization error is bounded with complementarity, whereas the algebraic error is computed very accurately with a novel and cheap technique.