Using adjoint solutions to estimate errors in global quantities

Adjoint solutions are widely used in computational electromagnetics to provide sensitivities of global quantities to variation in design parameters. They can also be used to provide estimates of the discretization error in these quantities. Electrostatic force and capacitance are considered. Results are obtained for two test problems, using different meshes and different polynomial orders of hierarchal, tetrahedral finite elements. The estimates track the true errors well over a wide range. Furthermore, the estimates are good enough that, when added to the computed quantities, they reduce the error in those quantities, often substantially.