Ladyzhenskaya–Babuška–Brezzi condition


In numerical partial differential equations, the Ladyzhenskaya–Babuška–Brezzi condition is a sufficient condition for a saddle point problem to have a unique solution that depends continuously on the input data. Saddle point problems arise in the discretization of Stokes flow and in the mixed finite element discretization of Poisson's equation. For positive-definite problems, like the unmixed formulation of the Poisson equation, most discretization schemes will converge to the true solution in the limit as the mesh is refined. For saddle point problems, however, many discretizations are unstable, giving rise to artifacts such as spurious oscillations. The LBB condition gives criteria for when a discretization of a saddle point problem is stable.

Saddle point problems

The abstract form of a saddle point problem can be expressed in terms of Hilbert spaces and bilinear forms. Let and be Hilbert spaces, and let, be bilinear forms.
Let, where, are the dual spaces. The saddle-point problem for the pair, is to find a pair of fields in, in such that, for all in and in,
For example, for the Stokes equations on a -dimensional domain, the fields are the velocity and pressure, which live in respectively the Sobolev space and the Lebesgue space.
The bilinear forms for this problem are
where is the viscosity.
Another example is the mixed Laplace equation where the fields are again the velocity and pressure, which live in respectively the spaces and.
Here, the bilinear forms for the problem are
where is the viscosity.

Statement of the theorem

Suppose that and are both continuous bilinear forms, and moreover that is coercive on the kernel of :
for all such that for all.
If satisfies the inf–sup or Ladyzhenskaya–Babuška–Brezzi condition
for all, then there exists a unique solution of the saddle-point problem.
Moreover, there exists a constant such that