The Mollification Method For Solving Several Cauchy Problems Of Elliptic Equations

The mollification method is an effective regularization method for solving ill-posed problems. In this paper, we will introduce the basic idea of solving ill-posed problems by using the mollification method. The construction of the mollification operators is also presented. Moreover we use the. mollification method with Gaussian kernel to solve several concrete inverse problems of elliptic PDE including the Cauchy problem of the Laplace equation, the Cauchy problem of the Helmholtz equation and the Cauchy problem of the modified Helmholtz equation. Convergence estimates between the exact solutions and their approximations are obtained. Numerical experiments are given to show the effectiveness of the mollification method.The analysis of the theory and the numerical results show that the mollification method is a stable, flexible, practical and effective regularization method.