Adaptive Uzawa algorithm for the Stokes equation

Based on the Uzawa algorithm, we consider an adaptive finite element method for the Stokes system. We prove linear convergence with optimal algebraic rates for the residual estimator (which is equivalent to the total error), if the arising linear systems are solved iteratively,e.g., by PCG. Our analysis avoids the use of discrete efficiency of the estimator. Unlike prior work, our adaptive Uzawa algorithm can thus avoid to discretize the given data and does not rely on an interior node property for the refinement.

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.

A relaxed non-linear inexact Uzawa algorithm for stokes problem

In this paper, we consider a Stokes problem arising in fluid dynamics and thermal science, which can be transformed to a symmetric saddle point problem by using the mixed finite element approximation. A relaxed non-linear inexact Uzawa algorithm is proposed for solving the problem, and the convergence of this algorithm is also considered. Numerical experiments are presented to show the efficiency of relaxed non-linear inexact Uzawa algorithm.

A special parameterized inexact Uzawa algorithm for symmetric saddle point problem

In this paper, we consider a symmetric saddle point problem arising in the fluid dynamics. A special parameterized inexact Uzawa algorithm is proposed for solving the symmetric saddle point problem. The convergence of this special algorithm is considered. Sufficient conditions for the convergence are given. Numerical experiments resulting from stokes problem are presented to show the efficiency of the algorithm.