A BDF2-semismooth Newton algorithm for the numerical solution of the Bingham flow with temperature dependent parameters

Abstract

This paper is devoted to the numerical solution of the non-isothermal instationary Bingham flow with temperature dependent parameters by semismooth Newton methods. We discuss the main theoretical aspects regarding this problem. Mainly, we discuss the existence of solutions for the problem, and focus on a multiplier formulation which leads us to a coupled system of PDEs involving a Navier–Stokes type equation and a parabolic energy PDE. Further, we propose a Huber regularization for this coupled system of partial differential equations, and we briefly discuss the well posedness of the regularized problem. A detailed finite element discretization, based on the so called (cross-grid P1) - Q0 elements, is proposed for the space variable, involving weighted stiffness and mass matrices. After discretization in space, a second order BDF method is used as a time advancing technique, leading, in each time iteration, to a nonsmooth system of equations, which is suitable to be solved by a semismooth Newton (SSN) algorithm. Therefore, we propose and discuss the main properties of a SSN algorithm, including the convergence properties. The paper finishes with two computational experiments that exhibit the main properties of the numerical approach.