Enhanced algorithms for solving the spectral discretization of the vorticity–velocity–pressure formulation of the Navier–Stokes problem

The objective of the article is to improve the algorithms for the resolution of the spectral discretization of the vorticity–velocity–pressure formulation of the Navier–Stokes problem in two and three domains. Two algorithms are proposed. The first one is based on the Uzawa method. In the second one we consider a modified global resolution. The two algorithms are implemented and their results are compared.

Krylov improvements of the Uzawa method for Stokes type operator matrices

The paper is devoted to Krylov type modifications of the Uzawa method on the operator level for the Stokes problem in order to accelerate convergence. First block preconditioners and their effect on convergence are studied. Then it is shown that a Krylov–Uzawa iteration produces superlinear convergence on smooth domains, and estimation is given on its speed.

An Optimal Finite Element Method with Uzawa Iteration for Stokes Equations including Corner Singularities

The Uzawa method is an iterative approach to find approximated solutions to the Stokes equations. This method solves velocity variables involving augmented Lagrangian operator and then updates pressure variable by Richardson update. In this paper, we construct a new version of the Uzawa method to find optimal numerical solutions of the Stokes equations including corner singularities. The proposed method is based on the dual singular function method which was developed for elliptic boundary value problems. We estimate the solvability of the proposed formulation and special orthogonality form for two singular functions. Numerical convergence tests are presented to verify our assertion.

A sensitivity study of relaxation parameter in Uzawa algorithm for the steady natural convection model

Purpose This paper aims to study the sensitivity of relaxation parameter in Uzawa method for the steady natural convection model. Design/methodology/approach Based on the continuous sensitivity equation method, associated sensitivity system is formed, which is solved by applying finite element method. Findings A decrease in sensitivity values of velocity, pressure and temperature is observed as the relaxation parameter increases. Originality/value The sensitivity study of relaxation parameter in Uzawa method is the first for the natural convection model, to the best of the authors’ knowledge. In fact, it is difficult to find an optimal relaxation parameter in the Uzawa method for the nonlinear problems. It is therefore important to understand how small changes in the relaxation parameter could affect the numerical solution of the nonlinear equations.