Thanks to a finite element method, we solve numerically parabolic partial differential equations on complex domains by avoiding the mesh generation, using a background regular mesh, not fitting exactly the domain and the real boundary of the domain like immersed boundary methods. Our technique follows the ϕ-FEM paradigm which suppose that the domain is given by a level-set function. The aim of the present paper is to prove a priori error estimates in L 2 (H 1) and L ∞ (L 2) norms for an implicit Euler discretization in time and to give numerical illustrations to highlight the performances of our technique. The advantage of our approach is that it combines optimal convergence accuracy, easy implementation process and fastness.