Abstract This technical note proves that, for a smooth vector optimization problem on a closed convex feasible set ordered by a pointed cone, the projected gradient direction depends continuously on the decision variable. Our argument is based on a simple and direct proof via a fixed-domain reformulation of the subproblem. We then give a necessary and sufficient dual characterization of this direction and show that its associated set-valued dual variable mapping is outer semicontinuous.

Abstract We analyze the existence of solutions for the finite-element discretized stationary and time-dependent sedimentation-consolidation partial differential equations (PDEs) using recent advances in the local-projection stabilization (LPS) method. The sedimentation model studied here couples gravitationally forced Stokes flow and a convection-diffusion equation for the solids concentration, with boundary conditions selected to mimic physical applications. This system is highly nonlinear and sensitive due to the non-constant velocity and concentration coupling terms, as well as the nonlinear flux term in the convection equation. We provide an overview of pre-existing methods already developed for solving this class of problem, establish the coercivity of the underlying operator, as well as conditions for the existence of discrete solutions. For convection-dominated regimes using non- inf-sup stable finite elements with LPS, we demonstrate that stabilization effectively eliminates interior and boundary layers in the velocity and pressure solutions. Numerical examples in two and three dimensions for the non-stationary case illustrate the theoretical results.