Bifurcation Problem

• Bifurcation Problem
• • Mass matrix

We refer to the docs of BifurcationKit.jl for an in-depth description of the bifurcation problems. Here, we only focus on the ones related to Gridap.

The main structure is GridapBifProblem which encodes a system of PDEs discretized with Gridap.jl. The weak form is written with the usual Gridap API, for example

res((u, v), p, (V1, V2)) = ∫( -p.D1 * ∇(u)⋅∇(V1) + NL1∘(u, v) ⋅ V1 +                              -p.D2 * ∇(v)⋅∇(V2) + NL2∘(u, v) ⋅ V2 ) * dΩ

and the problem is instantiated with

prob = GridapBifProblem(res, u0, par, V, U, dΩ, (@optic _.D1); jac = jac, mass = m0)

See the tutorials for complete examples.

Mass matrix

Many evolution PDEs write, after semi-discretization in space,

\[M\frac{\mathrm{d}z}{\mathrm{d}t} = F(z, p)\]

where $M$ is a mass matrix which can be singular. A typical example is the incompressible Navier–Stokes equations: the velocity has a time derivative but the pressure does not, so that the mass matrix has a zero block on the pressure dofs.

The stability of a stationary solution is obtained from the generalized eigenvalue problem

\[J\,\phi = \lambda\, M\,\phi\]

and not from the spectrum of $J$ alone (which contains spurious modes coming from the saddle-point structure). For this reason, GridapBifProblem is a subtype of BifurcationKit.AbstractDAEBifProblem and the mass matrix is used automatically by the DAE eigensolvers, see Eigen Solvers.

The mass matrix is assembled with

By default the L² mass $\int u\cdot v$ is used. For incompressible flows, one passes the velocity only mass, which produces the required zero block on the pressure:

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

The following BifurcationKit functions are specialized so that the mass matrix is correctly taken into account:

• BifurcationKit.is_mass_matrix_constant(prob) returns true;
• BifurcationKit.getmassmatrix(prob, x, p) returns get_mass_matrix(prob).