An anisotropic p-adaptation multigrid scheme for discontinuous galerkin methods

In this work, we present an anisotropic p-adaptation multigrid algorithm for discontinuous Galerkin methods for steady-state problems that uses a p-multigrid scheme both as a solver and as an anisotropic error estimator. To achieve this, we develop a new anisotropic truncation error estimator based on the tau-estimation method, that can be evaluated inside the multigrid cycle with a negligible extra cost. The new error estimator is cheaper to evaluate and more accurate than previous versions of the tau-estimation procedure. In our technique, a non-converged solution in a reference mesh is used to estimate the truncation error with the multigrid scheme for different combinations of polynomial orders in different directions inside every element, and the mesh is adapted accordingly to target a desired truncation error threshold. The accuracy and computational cost of the proposed p-anisotropic adaptation algorithm is tested for the steady viscous flow past a NACA0012 airfoil. A speed-up of 16 can be achieved in the proposed numerical example compared with the uniformly refined simulation without multigrid.