Motivated by the idea of having all the work chain parallelized for the Finite Element Method; we looked at the parallelization of the assembling phase on the FEM. We have used the FEniCSx collection of tools for solving Partial Differential Equations; used by the Simula laboratory for the FEM part, and we try to use the StarPU runtime for parallelizing manycore architectures developed by the STORM team at INRIA Bordeaux In this document, we explore different strategies for combining these tools.