In this note, we are concerned with the solvability of multifield coupled problems with different, often conflictual types of non-linearities. We bring into focus the challenges of getting EFM numerical solutions. As for instance, we share our investigations of the solvability of thixoviscoplastic flow problems in FEM settings. On one hand, nonlinear multifield coupled problems are often lacking unified FEM analysis due to the presence of different nonlinearities. Thus, the importance of treating auxiliary subproblems with different analysis tools to guarantee existence of solutions. Moreover, the nonlinear multifield problems are extremely sensitive to the coupling. On other hand, monolithic Newton-multigrid FEM solver shows a great success in getting numerical solutions for multifield coupled problems. Thixoviscoplastic flow problem is a perfect example in this regard. It is a two field coupled problem, by means of microstructure dependent plastic-viscosity as well as microstructure dependent yield stress, and microstructure and shear rate dependent buildup and breakdown functions. We adapt different numerical techniques to show the solvability of the problem, and expose the accuracy of FEM numerical solutions via the simulations of thixoviscoplastic flow problems in channel configuration.