FAQ

Which mass matrix should I use for the stability of an incompressible flow?

For the incompressible Navier–Stokes equations, the pressure has no time derivative, so the semi-discrete system is $M\dot z = F(z, p)$ with a singular mass matrix whose pressure block is zero. You should therefore pass the velocity only mass:

m0((u, p), (v, q)) = ∫( v ⊙ u ) * dΩprob = GridapBifProblem(res, u0, par, V, U, dΩ, lens; jac = jac, mass = m0)

Using the full L² mass (including $\int p\,q$) or the identity matrix changes the spectrum of the pencil and can spoil the detection of Hopf bifurcations. See Mass matrix.

Which eigensolver should I use for a differential-algebraic problem?

GridapBifProblem is a BifurcationKit.AbstractDAEBifProblem: whatever eigensolver you put in NewtonPar(eigsolver = ...) is automatically wrapped into EigenDAE, and the generalized eigenvalue problem $J\phi = \lambda M\phi$ is solved with the assembled mass matrix. You can also build it explicitly with EigenMassMatrix(M, eig). See Eigen Solvers.

The eigenvalues returned by my shift-invert eigensolver look wrong

With a shift-invert eigensolver (EigArpack, EigArnoldiMethod), the option which = LM() selects the eigenvalues closest to the shift sigma, not those with the largest real part. To monitor the stability, choose a shift near the imaginary axis where the bifurcation is expected (for example sigma ≈ iω for a Hopf point). See Eigen Solvers.