Abstract
The Cauchy problem of the Poisson equation in two spatial dimensions is considered in this paper, in which Dirichlet and Neumann data are simultaneously imposed on a part of the boundary of the domain. The problem can be regarded as a boundary inverse problem, in which the proper boundary conditions are to be indentified for the rest of the boundary. The problem can be reformulated as a minimization problem of a functional with constraints, which is minimized by the method of the steepest descent. The minimization problem is recast into successive solution of primary and dual boundary value problems for the Poisson equations. Some examples in this paper indicate that the numerical solutions are convergent and the scheme is confident.
