A High-order Dual Pressure Discontinuous Galerkin Method for Solving the Incompressible Navier-Stokes Equations

  • Elzaabalawy, Hashim (LHEEA )
  • Deng, Ganbo (LHEEA)
  • Visonneau, Michel (LHEEA)

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Solving the incompressible Navier-Stokes equations using high-order methods for large scale problems is not straightforward. In addition to the velocity-pressure coupling problem, the resulting linear system is badly conditioned due to the high-order discretization used. Most of the approaches in the literature are either not energy-stable or they result in extremely hard linear systems to solve. A novel method for solving the incompressible Navier-Stokes equations is presented based on the discontinuous Galerkin approach with the aim to be easily solved on modern hardware architecture. The method is constructed to be convenient with the multigrid approach while being energy-stable, exactly mass conserving, and momentum conserving. Additionally, the method can be solved in a matrix-free approach. The method relies on using dual pressure by defining the pressure inside the elements and on the faces in two different functional spaces similar to the technique used in hybridizable discontinuous Galerkin. While the velocity is approximated in the conventional space used in the discontinuous Galerkin method. The BR2 stabilization is used to deal with the diffusion term. The resulting linear system is solved using a \textit{p}-multigrid approach with a newly developed smoother that can deal with saddle point problems. The method showed potential to be successful for laminar flows. The method seems to be almost $p$ independent which is the main outcome of this method.