Borachok I. Numerical solution of the Cauchy problem for the Laplace equation in three dimensional double connected domains

The thesis is devoted to the numerical solution of the problem of reconstructing the Cauchy data of a harmonic function in a three-dimensional double connected domain. It is shown that the initial problem belongs to a class of ill-posed problems, in the sense of lack of stability on the input data. For a numerical solution of the initial problem, the direct Tikhonov regularization method and two iterative methods (the alternating method and the generalized Landveber method) are considered. In the case of Tikhonov's method, the Cauchy problem is reduced to an ill-posed system of two-dimensional integral equations by means of the theory of potential and using Green's formula. In both variants, the integral representation of the solution, the corresponding ill-posed system of integral equations, as well as the appearance of the desired Cauchy data on the internal boundary is given. For two systems of integral equations, the possibility of applying Tikhonov's regularization method has been investigated. The parametrization of the received systems of integral equations and the weak singularity in the kernel are investigated. By means of the discrete Galerkin projection method, the parametrized systems of integral equations are completely discretized, moreover unknown densities are approximated by a linear combination of spherical harmonics, and the integrals are approximated by means of corresponding cubature rules with superalgebraic order of convergence for analytic integral functions. To the obtained systems of linear equations, the Tikhonov regularization method was used, and the regularization parameter was chosen using the L-curve method. An iterative alternating method is given. This iteration method was proposed by V.Kozlov and V.Maz'ya for ill-posed problems with a self-adjoint operator. In this thesis, the idea of the method is extended to the case of three-dimensional domains. An algorithm for an iterative method is presented, which consists of sequential solving of two well-posed mixed Neumann-Dirichlet and Dirichlet-Neumann problems for the Laplace equation. The convergence and stability of this method are investigated. Two modifications of the generalized Landveber method for which no adjoint operator is required are constructed in the work. The algorithms of data of iterative procedures are presented, at each step of which it is necessary to solve the Dirichlet and Robin problem or the mixed Dirichlet-Neumann and Neumann-Dirichlet problems for the Laplace equation with the corresponding input data. The convergence and stability of the obtained iterative methods are investigated. The algorithms for the numerical solving of well-posed three-dimensional problems by an indirect method of integral equations are considered. A discrete Galerkin projection method is considered for the discretization of the obtained systems of the integral equations.