Some Ill-posed Problems For Elliptic And Parabolic Equations

In this thesis,we consider some ill-posed problems of linear elliptic and parabolic equations,i.e.,Backward parabolic equation in time and Cauchy problems of elliptic equation.For some difficult variable coefficient cases,several conditional stability results are given.Some regularization methods including Tikhonov regularization method,Cut-Off regularization method and Quasi-Boundary-Value method are used for these ill-posed problems respectively. A-priori choice rule for all regularization methods and a-posteriori choice rule for some of them are given.All corresponding error estimates are obtained.About the Quasi-Boundary-Value method,we give some properties after observing its applications. These properties are helpful for dealing with other ill-posed problems.In the numerical aspect,we use the finite difference method and the Fast Fourier Transform to implement all regularization methods for both the constant coefficient and variable coefficient cases.For the variable coefficient case of elliptic Cauchy problem,we consider its three dimensional numerical implementation since it is more ill-posed and the coefficient matrix of the linear system is huge.A Left-Preconditioned GMRES method is used.A good preconditioner is constructed and a fast direct solver is also given to make the GMRES method work well.The numerical results are consistent with the theoretical results.These results show that our regularization methods for these ill-posed problems work effectively.