The question that follows was phrased at JuliaCon:
given: suppose solving a 2D / 3D transient laminar Navier-Stokes problems (lid-driven cavity, flow part cylinder or airfoil) using Gridap, Trixi or similar
how do the following 2 options compare: a/ using linsolve within DiffEQ or b/ using TS/SNES/KPS/PC within PETSc ?
does is make sense to employ the DAE formulation to treat the pressure equation?
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.