Analysis of an HDG method for the Navier-Stokes equations with Dirac measures

In two dimensions, we analyze a hybridized discontinuous Galerkin (HDG) method for the Navier-Stokes equations with Dirac measures. The approximate velocity field obtained by the HDG method is shown to be pointwise divergence-free and $H$(div)-conforming. Under a smallness assumption on the continuous and discrete solutions, a posteriori error estimator, that provides an upper bound for the $L^2$-norm in the velocity, is proposed in the convex domain. In the polygonal domain, reliable and efficient a posteriori error estimator for the $W^{1,q}$-seminorm in the velocity and $L^q$-norm in the pressure is also proved. Finally, a Banach's fixed point iteration method and an adaptive HDG algorithm are introduced to solve the discrete system and show the performance of the obtained a posteriori error estimators.