[Paper Review] A Note on the Ladyzenskaja-Babuska-Brezzi Condition

Abner J. Salgado, Johnny Guzmán+1 more|arXiv (Cornell University)|Mar 8, 2012

Advanced Numerical Methods in Computational Mathematics11 references3 citations

TL;DR

This paper establishes the equivalence of the classical Ladyzenskaja-Babuška-Brezzi (LBB) condition and the generalized LBB (GLBB) condition for mixed finite element methods in the Stokes problem, under the assumption of quasi-uniform, shape-regular meshes. It proves that GLBB implies LBB when the domain has sufficient regularity, and further shows that GLBB implies a weighted inf–sup condition relevant to preconditioning and time-dependent problems.

ABSTRACT

The analysis of finite-element-like Galerkin discretization techniques for the stationary Stokes problem relies on the so-called LBB condition. In this work we discuss equivalent formulations of the LBB condition.

Motivation & Objective

• To clarify the relationship between the classical LBB condition and the generalized LBB (GLBB) condition in mixed finite element methods.
• To investigate whether GLBB implies the classical LBB condition on quasi-uniform meshes.
• To demonstrate that GLBB implies a weighted inf–sup condition used in preconditioning and time-dependent Stokes problems.
• To establish that GLBB is sufficient for the existence of an L²-bounded Fortin projection, ensuring stability and convergence.

Proposed method

• Uses the Scott-Zhang interpolation operator to construct a Fortin projection with L²-approximation properties.
• Applies inverse inequalities and mesh regularity assumptions (quasi-uniform, shape-regular) to control norms.
• Employs a weighted inf–sup condition involving the norm ‖·‖_H¹+ε⁻¹L² and ‖·‖_L²∩εH¹ to analyze preconditioning.
• Leverages the existence of a Fortin operator satisfying (2.8) to transfer continuous inf–sup stability to discrete settings.
• Uses case analysis based on ε relative to mesh size h to prove the weighted inf–sup condition.
• Relies on domain regularity (star-shaped with respect to a ball) to ensure continuous inf–sup stability.

Experimental results

Research questions

• RQ1 Is the generalized LBB condition (GLBB) equivalent to the classical LBB condition on quasi-uniform meshes?
• RQ2 Does the GLBB condition imply the weighted inf–sup condition used in preconditioning of time-dependent Stokes problems?
• RQ3 Can the existence of an L²-bounded Fortin projection be guaranteed under the GLBB condition?
• RQ4 Does GLBB imply LBB when the domain has H²-regularity for the Stokes solution?
• RQ5 Can GLBB be used to establish stability for all standard inf–sup stable finite element spaces, including Taylor-Hood and mini-elements?

Key findings

• On quasi-uniform, shape-regular meshes, the generalized LBB condition (GLBB) implies the classical LBB condition.
• When the domain has H²-regularity for the Stokes problem, GLBB is equivalent to LBB.
• The GLBB condition implies the weighted inf–sup condition (5.1) with a constant independent of ε and h.
• For the lowest-order Taylor-Hood element in 2D, GLBB implies the weighted inf–sup condition, and this extends to all Taylor-Hood elements in 2D and 3D.
• The GLBB condition ensures the existence of a Fortin projection that is bounded in L² and has optimal approximation properties.
• The results hold under the assumption of quasi-uniform meshes, and the authors leave open whether this condition can be removed.