Iterative methods for Stokes flow under nonlinear slip boundary condition coupled with the heat equation

Abstract

We study two iterative schemes for the finite element approximation of the heat equation coupled with Stokes flow under nonlinear slip boundary conditions of friction type. The iterative schemes are based on Uzawa's algorithm in which we decouple the computation of the velocity and pressure from that of the temperature by means of linearization. We derive some a priori estimates and prove convergence of these schemes. The theoretical results obtained are validated by means of numerical simulations.

Stokes equations under nonlinear slip boundary conditions coupled with the heat equation: A priori error analysis

“…Indeed, we demonstrate by means of simulations the reliability of the algorithm (in fact its discrete analog). A companion paper of this work has been accepted for publication in , and to the best of our knowledge these are the first contributions toward the mathematical understanding of fluid flow under nonlinear slip boundary condition coupled with the heat. The rest of the paper is organized as follows: Section 2 is concerned with the weak formulation, the construction of weak solution, and its finite element approximation. Section 3 is devoted to the convergence analysis of the finite element discretization. Section 4 is devoted to the formulation of the iterative scheme and its convergence analysis. Section 5 is devoted to numerical simulations, discussions, and conclusions. …”