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Ladyzhenskaya–Babuška–Brezzi condition

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In numerical partial differential equations, the Ladyzhenskaya–Babuška–Brezzi (LBB) 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.

The condition is variously referred to as the LBB condition, the Babuška–Brezzi condition, or the "inf-sup" condition.

Saddle point problems

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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 (in this context also sometimes called the Darcy equations) where the fields are again the velocity and pressure , which live in the spaces and , respectively. Here, the bilinear forms for the problem are

where is the inverse of the permeability tensor.

Statement of the theorem

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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 and for some , then there exists a unique solution of the saddle-point problem. Moreover, there exists a constant such that

The alternative name of the condition, the "inf-sup" condition, comes from the fact that by dividing by , one arrives at the statement

Since this has to hold for all and since the right hand side does not depend on , we can take the infimum over all on the left side and can rewrite the condition equivalently as

Connection to infinite-dimensional optimization problems

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Saddle point problems such as those shown above are frequently associated with infinite-dimensional optimization problems with constraints. For example, the Stokes equations result from minimizing the dissipation

subject to the incompressibility constraint

Using the usual approach to constrained optimization problems, one can form a Lagrangian

The optimality conditions (Karush-Kuhn-Tucker conditions) -- that is the first order necessary conditions—that correspond to this problem are then by variation of with regard to

and by variation of with regard to :

This is exactly the variational form of the Stokes equations shown above with

The inf-sup conditions can in this context then be understood as the infinite-dimensional equivalent of the constraint qualification (specifically, the LICQ) conditions necessary to guarantee that a minimizer of the constrained optimization problem also satisfies the first-order necessary conditions represented by the saddle point problem shown previously. In this context, the inf-sup conditions can be interpreted as saying that relative to the size of the space of state variables , the number of constraints (as represented by the size of the space of Lagrange multipliers ) must be sufficiently small. Alternatively, it can be seen as requiring that the size of the space of state variables must be sufficiently large compared to the size of the space of Lagrange multipliers .

References

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  • Boffi, Daniele; Brezzi, Franco; Fortin, Michel (2013). Mixed finite element methods and applications. Vol. 44. Springer.
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