Saddle-point problems and Stokes elements

Set V to the vector-valued zero-trace H¹ space and Q to square-integrable pressures with zero spatial mean. Use viscosity times the gradient inner product for a, and negative pressure times divergence for b. The velocity energy is coercive in the gradient norm. The continuous divergence inf-sup property on this connected domain gives the other stability ingredient. Pressure need only be L² in this weak formulation, so requiring its classical gradient pointwise would impose unnecessary regularity.

Adding a constant to pressure changes none of its pairings with zero-trace velocity tests, because the integral of their divergence vanishes. The mean-zero condition removes that gauge. On a disconnected domain one global mean condition would not remove all componentwise constant modes, so connectedness matters. Nonzero velocity or divergence data need appropriate flux compatibility and a lifting; the homogeneous full-Dirichlet setting here avoids silently introducing incompatible inflow or outflow.

For a conforming discrete pair, the matrix has a positive velocity block A and a rectangular discrete divergence block B. The zero pressure block records that pressure enforces a constraint rather than carrying its own positive energy. Eliminating velocity gives a pressure Schur complement B A inverse B transpose. Its positivity on the normalized pressure space depends on B detecting every remaining pressure direction. A pressure null mode makes this Schur complement singular even if A is perfectly well conditioned.

Uniform spectral control of a suitably norm-scaled Schur complement is closely related to the discrete inf-sup constant, rather than to its unscaled entries alone. This also guides block preconditioners: a pressure mass matrix may approximate the correct scale under a stable pair and the relevant viscosity convention, but that is an additional spectral statement. Counting more velocity coefficients than pressure coefficients cannot exclude a checkerboard mode or a stability margin that tends to zero under refinement.

Taylor–Hood uses continuous quadratic velocity and continuous linear pressure on simplices, with the appropriate boundary and pressure constraints. Its stability must be attached to an admissible mesh theorem. For example, the cited result in dimensions two and three assumes an exact polygonal or polyhedral domain with a fitted conforming shape-regular affine mesh family, and each simplex must have at least d internal edges. Those hypotheses are not supplied merely by choosing polynomial degrees. Other stable pairs and stabilizations have different assumptions and sometimes different discrete kernels.

Equal-order continuous linear velocity and pressure are not generally inf-sup stable without suitable modification, even though both variables have familiar approximation estimates. Conversely, eliminating the constant pressure gauge does not eliminate every possible spurious pressure mode. A verification should identify the exact spaces, physical norms, geometry, mesh family and boundary conditions, then supply uniform stability and approximation results. For an actual flow computation, algebraic solver accuracy and pressure normalization are additional checks after the variational pair has been justified.