Dr. Caucao-Paillán, Sergio

2020, Benavides, Gonzalo A., Caucao-Paillán, Sergio, Gatica, Gabriel N., Hopper, Alejandro A.

In this paper we introduce and analyze a new finite element method for a strongly coupled flow and transport problem in , , whose governing equations are given by a scalar nonlinear convection–diffusion equation coupled with the Stokes equations. The variational formulation for this model is obtained by applying a suitable dual-mixed method for the Stokes system and the usual primal procedure for the transport equation. In this way, and differently from the techniques previously developed for this and related coupled problems, no augmentation procedure needs to be incorporated now into the solvability analysis, which constitutes the main advantage of the present approach. The resulting continuous and discrete schemes, which involve the Cauchy fluid stress, the velocity of the fluid, and the concentration as the only unknowns, are then equivalently reformulated as fixed point operator equations. Consequently, the well-known Schauder, Banach, and Brouwer theorems, combined with Babuška–Brezzi’s theory in Banach spaces, monotone operator theory, regularity assumptions, and Sobolev imbedding theorems, allow to establish the corresponding well-posedness of them. In particular, Raviart–Thomas approximations of order for the stress, discontinuous piecewise polynomials of degree for the velocity, and continuous piecewise polynomials of degree for the concentration, becomes a feasible choice for the Galerkin scheme. Next, suitable Strang-type lemmas are employed to derive optimal a priori error estimates. Finally, several numerical results illustrating the performance of the mixed-primal scheme and confirming the theoretical rates of convergence, are provided.

2021, Caucao-Paillán, Sergio, Gatica, Gabriel N., Sandoval, Felipe

In this work we present and analyze a fully-mixed formulation for the nonlinear model given by the coupling of the Navier–Stokes and Darcy–Forchheimer equations with the Beavers–Joseph–Saffman condition on the interface. Our approach yields non-Hilbertian normed spaces and a twofold saddle point structure for the corresponding operator equation. Furthermore, since the convective term in the Navier–Stokes equation forces the velocity to live in a smaller space than usual, we augment the variational formulation with suitable Galerkin type terms. The resulting augmented scheme is then written equivalently as a fixed point equation, so that the well-known Schauder and Banach theorems, combined with classical results on nonlinear monotone operators, are applied to prove the unique solvability of the continuous and discrete systems. In particular, given an integer k ≥ 0, Raviart–Thomas spaces of order k, continuous piecewise polynomials of degree ≤k + 1 and piecewise polynomials of degree ≤k are employed in the fluid for approximating the pseudostress tensor, velocity and vorticity, respectively, whereas Raviart–Thomas spaces of order k and piecewise polynomials of degree ≤k for the velocity and pressure, constitute a feasible choice in the porous medium. A priori error estimates and associated rates of convergence are derived, and several numerical examples illustrating the good performance of the method are reported