Adjoint-based hp-adaptivity on anisotropic meshes for high-order compressible flow simulations

We present an efficient adjoint-based hp-adaptation methodology on anisotropic meshes for high order Discontinuous Galerkin schemes applied to (nonlinear) convection-diffusion problems, including the compressible Euler and Navier-Stokes equations. The refinement strategy is based on a recently proposed interpolation error estimate [Dolejší V. Anisotropic hp-adaptive method based on interpolation error estimates in the lq-norm. Applied Numerical Mathematics 2014;82:80-114.], as well as an adjoint-based error estimate. Using the two error estimates, we determine the size and the shape of the triangular mesh elements on the desired mesh to be used for the subsequent adaptation steps. This is done using the concept of mesh-metric duality, where metric tensors encode information about mesh elements. The metric tensors that correspond to the desired mesh elements can be passed to a metric-conforming mesh generator to create the required anisotropic mesh. The effectiveness of the adaptation methodology is demonstrated using numerical experiments, involving a scalar model problem with a strong boundary layer, as well as various compressible flow test cases.