So the thing is that Trixi.jl is fundamentally designed as a hyperbolic PDE solver, so the main scope are always phenomena with finite speed of information propagation.
That having said, stuff like Stokes Flow is really out of scope. For such scenarios I would forward you to Waterlily.jl, which tackle explicitly incompressible flows, thus probably more suited.
Back to the question:
In Trixi.jl, there is no need to solve a linear system due to the nature of the DG solver we employ.