Ladyzhenskaya–Babuška–Brezzi condition

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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