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Adaptive HDG Methods for the Steady-State Incompressible Navier–Stokes Equations

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Abstract

We consider a hybridizable discontinuous Galerkin method for the steady-state incompressible Navier–Stokes equations. We use polynomials of degree \(k+1\), k, k and k for approximations of the velocity, the velocity gradient, the pressure and the boundary traces. Some stability results for approximate solutions and some relationships between norms are provided. Moreover an a posteriori error estimator is introduced. By \(L^2\)-projection and inf-sup condition, we prove that the error estimator is robust for the global \(L^2\) errors in the velocity, the velocity gradient and the pressure. Finally, a Picard iteration method and an adaptive HDG algorithm are presented. Furthermore, several numerical examples are shown to validate the theoretical analysis.

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  1. School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, Guangdong, China

    Haitao Leng

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The work of Haitao Leng was supported by the Cultivation Project of SCNU (Grant No. 19KJ08) and the NSF of China (Grant No. 12001209).

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Leng, H. Adaptive HDG Methods for the Steady-State Incompressible Navier–Stokes Equations. J Sci Comput 87, 37 (2021). https://doi.org/10.1007/s10915-021-01456-5

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