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 ${\displaystyle V}$ and ${\displaystyle Q}$ be Hilbert spaces, and let ${\displaystyle a:V\times V\to \mathbb {R} }$, ${\displaystyle b:V\times Q\to \mathbb {R} }$ be bilinear forms. Let ${\displaystyle f\in V^{*}}$, ${\displaystyle g\in Q^{*}}$ where ${\displaystyle V^{*}}$, ${\displaystyle Q^{*}}$ are the dual spaces. The saddle-point problem for the pair ${\displaystyle a}$, ${\displaystyle b}$ is to find a pair of fields ${\displaystyle u}$ in ${\displaystyle V}$, ${\displaystyle p}$ in ${\displaystyle Q}$ such that, for all ${\displaystyle v}$ in ${\displaystyle V}$ and ${\displaystyle q}$ in ${\displaystyle Q}$,

${\displaystyle {\begin{aligned}a(u,v)+b(v,p)&=\langle f,v\rangle \\b(u,q)&=\langle g,q\rangle .\end{aligned}}}$

For example, for the Stokes equations on a ${\displaystyle d}$-dimensional domain ${\displaystyle \Omega }$, the fields are the velocity ${\displaystyle u}$ and pressure ${\displaystyle p}$, which live in respectively the Sobolev space ${\displaystyle H^{1}(\Omega )^{d}}$ and the Lebesgue space ${\displaystyle L^{2}(\Omega )}$. The bilinear forms for this problem are

${\displaystyle {\begin{aligned}a(u,v)&=\int _{\Omega }\mu \nabla u:\nabla v\,dx\\b(u,q)&=\int _{\Omega }(\nabla \cdot u)q\,dx,\end{aligned}}}$

where ${\displaystyle \mu }$ is the viscosity.

Another example is the mixed Laplace equation (also called the Darcy equation) where the fields are again the velocity ${\displaystyle u}$ and pressure ${\displaystyle p}$, which live in respectively the spaces ${\displaystyle H_{\text{div}}(\Omega )^{d}}$ and ${\displaystyle L^{2}(\Omega )}$. Here, the bilinear forms for the problem are

${\displaystyle {\begin{aligned}a(u,v)&=\int _{\Omega }u\cdot v\,dx\\b(u,q)&=\int _{\Omega }(\nabla \cdot u)q\,dx,\end{aligned}}}$

where ${\displaystyle \mu }$ is the viscosity.

Statement of the theorem

Suppose that ${\displaystyle a}$ and ${\displaystyle b}$ are both continuous bilinear forms, and moreover that ${\displaystyle a}$ is coercive on the kernel of ${\displaystyle b}$:

${\displaystyle a(v,v)\geq \alpha \|v\|_{V}^{2}}$

for all ${\displaystyle v}$ such that ${\displaystyle b(v,q)=0}$ for all ${\displaystyle q\in Q}$. If ${\displaystyle b}$ satisfies the inf–sup or Ladyzhenskaya–Babuška–Brezzi condition

${\displaystyle \sup _{v\in V,v\neq 0}{\frac {b(v,q)}{\|v\|_{V}}}\geq \beta \|q\|_{Q}}$

for all ${\displaystyle q}$, then there exists a unique solution ${\displaystyle u,p}$ of the saddle-point problem. Moreover, there exists a constant ${\displaystyle C}$ such that

${\displaystyle \|u\|_{V}+\|p\|_{Q}\leq C(\|f\|_{V^{*}}+\|g\|_{Q^{*}}).}$

References

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

External links

• A Mixed Variational Formulation for 3D Linear and Nonlinear Magnetostatics
• Advanced finite element methods lecture notes
• Inf-sup condition and locking: understanding and circumventing