Banach spaces-based mixed finite element methods for a steady sedimentation-consolidation system

Abstract

We introduce and analyze two Banach spaces-based new mixed finite element methods for a flow and transport model commonly encountered in sedimentationconsolidation processes, whose governing equations are given by the Brinkman flow with variable viscosity, coupled with a nonlinear advection–diffusion equation. The first variational formulation is based on a mixed approach for the Brinkman problem (written in terms of Cauchy stress and bulk velocity of the mixture) and the usual primal weak form for the transport equation. In turn, the second variational formulation arises from the introduction of the gradient of the solids volume fraction and the total (diffusive plus advective) flux for the concentration as new unknowns, which yields a momentum-conserving fully–mixed approach as the resulting system of equations. The respective continuous and discrete formulations are equivalently reformulated as fixed-point operator equations, whose solvability is established by combining the Schauder, Banach, and Brouwer theorems, with, among others, the Babuška–Brezzi theory and a recently introduced theory for perturbed saddle-point problems, both in Banach spaces, along with suitable regularity assumptions, Sobolev embeddings, and Rellich–Kondrachov compactness theorems. The mixed–primal and fully-mixed Galerkin schemes employ the classical Raviart–Thomas, piecewise continuous, and piecewise discontinuous polynomial approximations, for their corresponding unknowns. Next, Strang-type inequalities are utilized to rigorously derive optimal error estimates in the natural norms, which, combined with the approximation properties of the chosen finite element spaces yield optimal rates of convergence with respect to the mesh size. Finally, several numerical results illustrating the performance of both schemes and confirming the theoretical convergence rates are presented.