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Thanks to a finite element method, we solve numerically parabolic partialdifferential equations on complex domains by avoiding the mesh generation,using a regular background mesh, not fitting the domain and its real boundaryexactly. Our technique follows the phi-FEM paradigm, which supposes that thedomain is given by a level-set function. In this paper, we prove a priori errorestimates in l2(H1) and linf(L2) norms for an implicit Euler discretization intime. We give numerical illustrations to highlight the performances of phi-FEM,which combines optimal convergence accuracy, easy implementation process andfastness.

phi-FEM for the heat equation: optimal convergence on unfitted meshes in space