2015/06/24 by Weifeng Qiu, Ke Shi, Qiu, Weifeng +1
Engineering · #Advanced Numerical Methods in Computational Mathematics #Computational Fluid Dynamics and Aerodynamics #Electromagnetic Simulation and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.1506.07543
openalex publication_date 2015/06/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We present a superconvergent hybridizable discontinuous Galerkin (HDG) method for the steady-state incompressible Navier-Stokes equations on general polyhedral meshes. For arbitrary conforming polyhedral mesh, we use polynomials of degree k+1, k, k to approximate the velocity, velocity gradient and pressure, respectively. In contrast, we only use polynomials of degree k to approximate the numerical trace of the velocity on the interfaces. Since the numerical trace of the velocity field is the only globally coupled unknown, this scheme allows a very efficient implementation of the method. For the stationary case, and under the usual smallness condition for the source term, we prove that the method is well defined and that the global L2-norm of the error in each of the above-mentioned variables and the discrete H1-norm of the error in the velocity converge with the order of k+1 for k>=0. We also show that for k>=1, the global L2-norm of the error in velocity converges with the order of k+2. From the point of view of degrees of freedom of the globally coupled unknown: numerical trace, this method achieves optimal convergence for all the above-mentioned variables in L2-norm for k>=0, superconvergence for the velocity in the discrete H1-norm without postprocessing for k>=0, and superconvergence for the velocity in L2-norm without postprocessing for k>=1.