A posteriori error analysis for the Navier-Stokes equations with non-smooth data
We study the stationary Navier-Stokes equations with Dirichlet boundary data in L2, a setting in which the limited regularity of the solution prevents the direct application of standard a posteriori error estimation techniques. To address this issue, we introduce a regularized formulation that yields a well-posed approximation of the original problem and admits a conforming finite element discretization. Using Taylor-Hood P2P1 elements, we construct a residual-based a posteriori error estimator and establish its reliability and efficiency under suitable smallness assumptions on the data. We derive computable upper and lower bounds in an appropriate norm that relate the estimator to the error between the exact solution of the original Navier-Stokes problem and its finite element approximation, showing that the estimator accurately reflects the finite element error. These results provide a rigorous foundation for the analysis and implementation of adaptive finite element methods for incompressible flows with low-regularity Dirichlet boundary data.