My work studies task-based runtime support for finite element method (FEM) assembly using StarPU. The idea is to replace bulk-synchronous loops with parallelized approach. Local kernels were rewritten in modern C++20/23, relying on non-owning views (std::span, std::mdspan) and a preallocated argument table to avoid repeated allocations. I also used METIS for partitioning and a greedy coloring step to guarantee conflict-free execution.The implementation is done for the project DOLFINx/FEniCS. On the 2D Poisson problem, assembly is already cheap and preprocessing dominates, so there is little to no gain and even loss of time on small model. On a 3D hyperelasticity benchmark, where each element is more expensive, the task-based model works well. Grouping cells into partitions reduces the number of tasks and keeps scheduling overhead low while maintaining balance. These results suggest that task-based assembly is a practical choice for nonlinear or compute-intensive models. Future extensions will need to focus on NUMA-aware scheduling, custom StarPU schedulers, and GPU support.