Today, topology optimization is a key tool in the mechanical, automotive, and aerospace industries. It has seen significant development since its introduction to solid mechanics applications in 1988 and has expanded to other disciplines, such as acoustics, fluid mechanics, and heat transfer. The fundamental idea is to distribute material within a predefined domain by minimizing a chosen objective and meeting a set of constraints, such as minimizing power dissipation or maximizing heat transfer subject to a limited fluid volume or amount of solid material. The process involves repeated system analyzes, gradient evaluation steps using adjoint sensitivity analysis, and design updates based on mathematical programming methods. The result of the topology optimization procedure is a bitmap image of the design. The method's ability to modify every pixel/voxel results in design freedom unmatched by any other approach. However, this freedom comes at the cost of substantially increasing the required computational power. Finer discretizations provide better design descriptions while demanding the use of large parallel machines. Furthermore, the size of the discrete problems necessitates the deployment of iterative techniques. However, the sharp transition between different physics (solid/fluid) combined with heat transfer results in ill-conditioned problems and a considerable number of iterations. Thus, the talk compares techniques for solving steady-state linearized Navier-Stokes and Stokes equations penalized with additional Brinkman terms. The results will be demonstrated on 2D and 3D fluid flow topology optimization problems with target outflow and power dissipation objectives and constraints implemented using the MFEM library.