Dr. Poza-Diaz, Abner

For the Stokes–Darcy coupled problem, which models a fluid that flows from a free medium into a porous medium, we introduce and analyze an adaptive stabilized finite element method using Lagrange equal order element to approximate the velocity and pressure of the fluid. The interface conditions between the free medium and the porous medium are given by mass conservation, the balance of normal forces, and the Beavers–Joseph–Saffman conditions. We prove the well-posedness of the discrete problem and present a convergence analysis with optimal error estimates in natural norms. Next, we introduce and analyze a residual-based a posteriori error estimator for the stabilized scheme. Finally, we present numerical examples to demonstrate the performance and effectiveness of our scheme.

This work aims to introduce and analyze an adaptive stabilized finite element method to solve a nonlinear Darcy equation with a pressure-dependent viscosity and mixed boundary conditions. We stated the discrete problem’s well-posedness and optimal error estimates, in natural norms, under standard assumptions. Next, we introduce and analyze a residual-based a posteriori error estimator for the stabilized scheme. Finally, we present some two- and three-dimensional numerical examples which confirm our theoretical results.

In this brief note, we extend the initial investigation proposed in John et al. (2016) to study the Stokes equations with variable viscosity considering a stabilized finite element scheme based on equal order polynomials for approximating the velocity and the pressure. We establish optimal a priori error estimates for this new stabilized scheme, which are further confirmed by numerical tests. Moreover, these tests show a weak viscosity dependence in the error bounds for different orders of magnitude between 𝜈max and 𝜈min.