Analysis of Iterative Methods for the Steady and Unsteady Stokes Problem: Application to Spectral Element Discretizations

A new and detailed analysis of the basic Uzawa algorithm for decoupling of the pressure and the velocity in the steady and unsteady Stokes operator is presented. The paper focuses on the following new aspects: explicit construction of the Uzawa pressure-operator spectrum for a semiperiodic model problem; general relationship of the convergence rate of the Uzawa procedure to classical inf-sup discretization analysis; and application of the method to high-order variational discretization

An explicit/implicit Runge–Kutta-based PFEM model for the simulation of thermally coupled incompressible flows

The final publication is available at Springer via http://dx.doi.org/10.1007/s40571-019-00229-0A semi-explicit Lagrangian scheme for the simulation of thermally coupled incompressible flow problems is presented. The model relies on combining an explicit multi-step solver for the momentum equation with an implicit heat equation solver. Computational cost of the model is reduced via application of an efficient strategy adopted for the solution of momentum/continuity system by the authors in their previous work. The applicability of the method to solving thermo-mechanical problems is studied via various numerical examples.Peer ReviewedPostprint (author's final draft

An efficient split-step framework for non-Newtonian incompressible flow problems with consistent pressure boundary conditions

Incompressible flow problems with nonlinear viscosity, as they often appear in biomedical and industrial applications, impose several numerical challenges related to regularity requirements, boundary conditions, matrix preconditioning, among other aspects. In particular, standard split-step or projection schemes decoupling velocity and pressure are not as efficient for generalised Newtonian fluids, since the additional terms due to the non-zero viscosity gradient couple all velocity components again. Moreover, classical pressure correction methods are not consistent with the non-Newtonian setting, which can cause numerical artifacts such as spurious pressure boundary layers. Although consistent reformulations have been recently developed, the additional projection steps needed for the viscous stress tensor incur considerable computational overhead. In this work, we present a new time-splitting framework that handles such important issues, leading to an efficient and accurate numerical tool. Two key factors for achieving this are an appropriate explicit–implicit treatment of the viscous and convective nonlinearities, as well as the derivation of a pressure Poisson problem with fully consistent boundary conditions and finite-element-suitable regularity requirements. We present first- and higher-order stepping schemes tailored for this purpose, as well as various numerical examples showcasing the stability, accuracy and efficiency of the proposed framework

An explicit/implicit Runge–Kutta-based PFEM model for the simulation of thermally coupled incompressible flows

A semi-explicit Lagrangian scheme for the simulation of thermally coupled incompressible flow problems is presented. The model relies on combining an explicit multi-step solver for the momentum equation with an implicit heat equation solver. Computational cost of the model is reduced via application of an efficient strategy adopted for the solution of momentum/continuity system by the authors in their previous work. The applicability of the method to solving thermo-mechanical problems is studied via various numerical examples