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. The abstract form of a saddle point problem can be expressed in terms of Hilbert spaces and bilinear forms. Let formula_1 and formula_2 be Hilbert spaces, and let formula_3, formula_4 be bilinear forms. Let formula_5, formula_6 where formula_7, formula_8 are the dual spaces. The saddle-point problem for the pair formula_9, formula_10 is to find a pair of fields  formula_11 in formula_1, formula_13 in formula_2 such that, for all formula_15 in formula_1 and formula_17 in formula_2, For example, for the Stokes equations on a formula_20-dimensional domain formula_21, the fields are the velocity formula_11 and pressure formula_13, which live in respectively the Sobolev space formula_24 and the Lebesgue space formula_25. The bilinear forms for this problem are where formula_27 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 formula_11 and pressure formula_13, which live in the spaces formula_30 and formula_25, respectively. Here , the bilinear forms for the problem are where formula_33 is the inverse of the permeability tensor. Statement of the theorem. Suppose that formula_9 and formula_10 are both continuous bilinear forms, and moreover that formula_9 is coercive on the kernel of formula_10: for all formula_15 such that formula_40 for all formula_41. If formula_10 satisfies the inf–sup or Ladyzhenskaya–Babuška–Brezzi condition for all formula_17 and for some formula_45, then there exists a unique solution formula_46 of the saddle-point problem. Moreover, there exists a constant formula_47 such that The alternative name of the condition, the "inf-sup" condition, comes from  the fact that by dividing by formula_49, one arrives at the statement Since this has to hold for all formula_41 and since the right hand side does not depend on formula_17, we can take the infimum over all formula_17 on the left side and can rewrite the condition equivalently as Connection to infinite-dimensional optimization problems. 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 formula_58 with regard to formula_11 and by variation of formula_61 with regard to formula_62: This is exactly the variational form of the Stokes equations shown above with The inf-sup conditions can in this context then be understand 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 formula_1 of state variables formula_11, the number of constraints (as represented by the size of the space formula_2 of Lagrange multipliers formula_62) must be sufficiently small. Alternatively, it can be seen as requiring that the size of the space formula_1 of state variables formula_11 must be sufficiently large compared to the size of the space formula_2 of Lagrange multipliers formula_62.