Effectivity of Output-Based Error Estimates for High-Order Aerodynamics Simulations

Output-based error estimation allows for the quantification of the effect of discretization errors on scalar outputs, and the localization of these errors for adaptive refinement. Key to the estimate is the adjoint-weighted residual (AWR), which at the discrete level is an inner product between an adjoint vector and a residual vector. Although the AWR is a simple operation, the effectivity of the resulting error estimate depends on choices made in the implementation, such as the construction of the fine space, and on the discretization, including boundary conditions. This paper investigates the error estimate effectivity for canonical flows in aerodynamics, comparing estimated errors to the actual errors computed relative to high-resolution simulations. One particular discretization is of interest: a high-order discontinuous finite-element method, which allows for both variable order and mesh refinement, in this case on unstructured meshes. The study is performed in an adaptive setting, since error estimation and adaptation are intrinsically tied together: error estimates improve with mesh quality, the generation of which depends on the accuracy of the error estimates. The results show that the effectivity is sensitive to the implementation choices, particularly on problems of increasing complexity, up to the Reynolds-averaged Navier-Stokes simulations, but that proper choices yield effective estimates. Aspects of the discretization that affect the error estimates are highlighted, including choice of fine space, order-dependence of the discrete residuals, viscous stabilization, and wall-distance representation.