Accelerated Dirichlet-Robin Alternating Algorithm for Solving the Cauchy Problem for an Elliptic Equation using Krylov Subspaces

In this thesis, we study the Cauchy problem for an elliptic equation. We use Dirichlet-Robin iterations for solving the Cauchy problem. This allows us to include in our consideration elliptic equations with variable coefficient as well as Helmholtz type equations. The algorithm consists of solving mixed boundary value problems, which include the Dirichlet and Robin boundary conditions. Convergence is achieved by choice of parameters in the Robin conditions. We have also reformulated the Cauchy problem for the Helmholtz equation as an operator equation. We investigate the conditions under which this operator equation is well-defined. Furthermore, we have also discussed possible extensions to the case where the Helmholtz operator is replaced by non-symmetric differential operators by using similar operator equations and model problems which are used for symmetric differential operators. We have observed that the Dirichlet - Robin iterations are equivalent to the classical Landweber iterations. Having formulated the problem in terms of an operator equation is an advantage since it lets us to implement more sophisticated iterative methods based on Krylov subspaces. In particular, we consider the Conjugate gradient method (CG) and the Generalized minimal residual method (GMRES). The numerical results shows that all the methods work well.