This paper is devoted to the mathematical analysis of a method based on fictitious domain approach. Boundary conditions of Robin type (also known as Fourier boundary conditions) are enforced using a penalization method. A complete description of the method and a full analysis are provided for univariate elliptic and parabolic problems using finite difference approximation. Numerical evidence of the predicted estimations is provided as well as numerical results for a nonlinear problem and a first extension of the method in the bivariate situation is proposed.