Numerical Methods For Linear Systems From Fictitious Domain Methods

Numerical method for saddle-point systems is a hot subject in the context of scientific computing,which is of great importance in theory and application.For example,mixed element methods, widely used in engineering applications,are the important sources of saddle-point problems.Due to the indefiniteness and poor spectral property of the coefficient matrix,it is very difficult and challenging to propose efficient numerical methods tor such problems.Fictitious domain method is a class of effective numerical methods tor solving partial differential equations.The method first rewrite the original problem defined on an irregular domain to an equivalent problem defined on a regular domain covering the original one,then discrete the new problem by the mixed element method,and finally obtains a saddle-point problem.Therefore,it is very important to provide efficient numerical methods for the corresponding saddle-point problem,in order to promote significantly the total performance of the fictitious domain method.In this thesis,with the fictitious domain method for solving second-order elliptic problems as a typical example,a systematic study is developed tbr numerically solving the corresponding linear systems.The methods include Parameterized Inexact Uzawa method,Preconditioned Uzawa method(PIU),Uzawa method, GMRES method,Preconditioned GMRES method.The numerical results show that the PIU method is an effective method when the iterative parameters are given appropriately.The Preconditioned GMRES method also performs very well.Moreover,a new simplified proof is also given to verify the unitbrm stability of the underlying mixed element method.