Optimization Of Some Uzawa Type Methods For Saddle Point Problems

Saddle point problem aries in many applications,such as computational fluid dynamics,approximation theory,domain decomposition algorithm.Thus the numerical techniques for saddle points are of great importance for scientific and engineering computations.Because of the large scale of the underlying problem,the iterative methods become more and more prevalent in this field.However,in an iterative method,the choice of the relaxation parameter affects dramatically its performance.In this thesis,we taking the Stokes equation as an example,investigate at continuous level by Fourier analysis how to choose the relaxation parameter in an iterative method.We obtain the following results: 1.We derived the optimal relaxation parameter for the classical Uzawa method;2.For a class of modified generalized parameterized inexact Uzawa algorithm see [ F.Chen,Y.-L.Jiang,A generalization of the inexact parameterized Uzawa methods for saddle point problems,Applied Mathematics and Computation 206(2)(2008)765–771.]),we obtain the optimal relaxation parameter and the corresponding convergence rate estimate.We find that the optimal relaxation parameter and the corresponding convergence rate are functions of the viscosity ?.Thus,we should take the viscosity into account when design a numerical algorithm for Stokes equation,though it could be scaled out from the equation first.3.Based on a generalized inexact Uzawa algorithm,we propose a block preconditioner for saddle point problem.At last,we use numerical examples to validate our theoretical findings.