A generalized framework for high order anisotropic mesh adaptation

We present a method for anisotropically adapting meshes for high order (p>2) finite volume methods. We accomplish this by assuming a polynomial error measure based on the local reconstruction. We then use Fourier series to choose a metric function that approximates our error measure. This approach is theoretically valid for any solution order, with any number of variables and any number of dimensions. Using both second and third order solvers, we present examples of two-dimensional subsonic viscous and inviscid flow around the NACA-0012 airfoil. These test cases demonstrate that mesh refinement based on our metric converges to an accurate solution much faster than with uniform refinement. For subsonic viscous flows, we also show that anisotropic refinement is more efficient than isotropic refinement. By limiting the magnitude of our error measure for a transonic inviscid flow, we demonstrate that our metric can be effective even in the presence of discontinuities. We also examined using second derivatives to estimate the error for third order solutions. While this is less theoretically sound, we found little difference in the resulting accuracy.