The method of invariant embedding for the solutions of boundary value problems yields an equivalent formulation to the initial boundary value problems by a system of Riccati operator differential equations. A combined technique based on invariant embedding approach and Yosida regularization is proposed in this paper for solving abstract Riccati problems and Dirichlet problems for the Poisson equation over a circular domain. We exhibit, in polar coordinates, the associated Neumann to Dirichlet operator, somme concrete properties of this operator are given. It also comes that from the existence of a solution for the corresponding Riccati equation, the problem can be solved in appropriate Sobolev spaces.