Hierarchical basis a posteriori error estimates for compressible Stokes flows Original Research Article

In this paper we study the a posterior error estimates for the barotropic linear and nonlinear compressible viscous Navier–Stokes system on the use of hierarchical basis functions. An abstract formulation for the linear system is proposed in (Kellogg and Liu, 1993), and using it, it is shown that the analysis given in (Bank and Smith, 1993) can be applied and generalized to the abstract formulation. As examples we apply the theory to the barotropic compressible viscous Navier–Stokes equations (Kellogg and Liu, 1993; Beirao Da Veiga, 1987), which are of mixed type, i.e., elliptic in momentum equations and hyperbolic in continuity one. Consequently, it is shown that the defined a posteriori error estimates for the linear and nonlinear problems are bounded below and above by the true error in different norms (see (1.1), (1.2)). As for this, one can imagine that the convective term of the continuity equation may degrade the error bounds but it is not known if this is in fact the case, or if it is an artifact of the formulation.