Hybridizable Discontinuous Galerkin with degree adaptivity for the incompressible Navier–Stokes equations

• First p-adaptive strategy for incompressible Navier–Stokes with Hybridizable Discontinuous Galerkin.
• Simple HDG error estimator allows controlling engineering outputs such as drag and lift.
• Results show efficiency of proposed approach for steady state and transient simulations.

A degree adaptive Hybridizable Discontinuous Galerkin (HDG) method for the solution of the incompressible Navier–Stokes equations is presented. The key ingredient is an accurate and computationally inexpensive a posteriori error estimator based on the super-convergence properties of HDG. The error estimator drives the local modification of the approximation degree in the elements and faces of the mesh, aimed at obtaining a uniform error distribution below a user-given tolerance in a given output of interest. Three 2D numerical examples are presented. High efficiency of the proposed error estimator is found, and an important reduction of the computational effort is shown with respect to non-adaptive computations, both for steady state and transient simulations.