An optimization-based framework for anisotropic hp-adaptation of high-order discretizations

Abstract

This paper presents a novel framework for hp-adaptation of high-order discontinuous finite element discretizations for compressible flow simulation. Using the sensitivities of an adjoint-based error estimate, our method seeks element size, shape, and polynomial degree distributions which minimize an adjoint-based error estimate for a specified number of degrees of freedom. This approach results in an optimized hp-mesh tailored to yield the most accurate prediction of an output quantity of interest, such as aerodynamic coefficients, at a given computational cost. The proposed approach features a reduced dependence on user-defined parameters compared to established fixed-fraction adjoint-based adaptive methods. It provides a unifying framework where adaptation decisions such as isotropic/anisotropic, h-/p-refinement/coarsening do not only rely on local arbitrary measures of solution anisotropy and smoothness, but rather where a globally optimal distribution of degrees of freedom is sought to minimize the error in the chosen quantity of interest.