7 Pressure Stability, Inf–Sup, and Stabilization
7.1 The discrete compatibility question
For many mixed incompressible-flow formulations, the finite-dimensional velocity and pressure spaces must satisfy a discrete inf–sup (Ladyzhenskaya–Babuška–Brezzi) condition, expressed schematically as
\[\inf_{q_h\ne0}\sup_{\mathbf v_h\ne0}\frac{b(\mathbf v_h,q_h)}{\lVert\mathbf v_h\rVert_V\lVert q_h\rVert_Q}\ge\beta>0,\]
with a constant \(\beta\) that should remain bounded away from zero as the mesh is refined for the family under consideration.
7.2 What instability can look like
An unstable choice of spaces may create pressure oscillations or unreliable pressure fields even if a linear solver reports a very small algebraic residual. Refining an unsuitable discretization does not automatically cure the underlying compatibility problem.
7.3 Common approaches
Use a stable pair of spaces, such as suitable Taylor–Hood elements, or a correctly designed stabilized formulation. Stabilized equal-order methods add terms that require their own consistency and parameter analyses.
7.4 Separate different kinds of stability
- PDE stability: Behavior of the continuous mathematical problem.
- Discrete stability: Compatibility of finite-element spaces and formulation.
- Time-integration stability: Restrictions associated with the time-stepping method.
- Solver robustness: Conditioning, preconditioning, and convergence of the algebraic solve.
These are related but not interchangeable.
Living chapter. This is an initial working treatment. Derivations, examples, and references will be expanded in future editions.