As numerical simulations are applied to more complex and large-scale problems, solution verification becomes increasingly important in ensuring the accuracy of the computed results. Although improvements in computer hardware have brought expensive simulations within reach, efficiency is still paramount, especially in the context of design optimization and uncertainty quantification. This work addresses both of these needs through contributions to solution-based adaptive algorithms, in which the discretization is modified through feedback of solution error estimates so as to improve the accuracy.  In particular, new methods are developed for two discretizations relevant to Computational Fluid Dynamics: the Active Flux method and the discontinuous Galerkin method. For the Active Flux method, which is fully-discrete third-order discretization, both the discrete and continuous adjoint methods are derived and used to drive mesh (h) refinement and dynamic node movement, also known as r-adaptation. For the discontinuous Galerkin method, which is an arbitrary-order finite-element discretization, efficiency improvements are presented for computing and using error estimates derived from the discrete adjoint, and a new r-adaptation strategy is presented for unsteady problems. For both discretizations, error estimate efficacy and adaptive efficiency improvements are shown relative to other strategies. Additionally, an automated one-to-one non-conformal mesh interface creation process will be introduced. This automated non-conformal mesh interface creation process eliminates human setup error while reducing the case setup time by 50% on average, for practical industrial applications.