  In this work, we derive two-sided a posteriori error estimates for the
dual-weighted residual (DWR) method. We consider both single and multiple goal
functionals. Using a saturation assumption, we derive lower bounds yielding the
efficiency of the error estimator. These results hold true for both nonlinear
partial differential equations and nonlinear functionals of interest.
Furthermore, the DWR method employed in this work accounts for balancing the
discretization error with the nonlinear iteration error. We also perform
careful studies of the remainder term that is usually neglected. Based on these
theoretical investigations, several algorithms are designed. Our theoretical
findings and algorithmic developments are substantiated with some numerical
tests.
