InfSupStability

Published configurations

Measures the finite-mesh stability of the Stokes velocity–pressure coupling through independently checked spectral calculations.

A mixed discretization must control every nonconstant pressure mode through the discrete divergence pairing. The finite-mesh stability constant is \(\beta_h=\inf_{q_h\ne0}\sup_{v_h\ne0}|b(v_h,q_h)|/(\|v_h\|_V\|q_h\|_Q)\) on the zero-mean pressure and essential-boundary velocity spaces.

Discretizations

Velocity degrees two, three and four are paired with pressure degree one lower on six cell geometries, affine and quadratic maps, several mesh resolutions, and quadrature orders sixteen and twenty. Velocity gradient energy, divergence and pressure mass matrices are assembled before timing, with unit viscosity in the energy convention.

Generalized pressure spectrum

Essential velocity degrees of freedom are removed. If \(c\) represents constant pressure, a basis \(T\) satisfying \(T^{\mathsf T}Mc=0\) defines \(M_0=T^{\mathsf T}MT\) and \(S_0=T^{\mathsf T}BA^{-1}B^{\mathsf T}T\). The generalized spectrum solves \(S_0z_i=\lambda_iM_0z_i\); its smallest eigenvalue gives \(\beta_h^2\) for the stated velocity and pressure norms.

Generalized Schur spectrum

Sparse LDLT factorization and blocked solves form the pressure-sized Schur operator without explicitly computing an inverse. A generalized self-adjoint eigensolver obtains its spectrum in the pressure mass metric.

Independent singular-value route

Independent Cholesky factorization and pressure whitening give \(D=C^{-1}T^{\mathsf T}BP^{\mathsf T}L^{-\mathsf T}\), where \(PAP^{\mathsf T}=LL^{\mathsf T}\) and \(M_0=CC^{\mathsf T}\). Incremental Givens QR retains a pressure-sized triangular factor for the SVD; squared singular values are compared with the Schur eigenvalues. The full dense velocity–pressure coupling is not retained.

Measurement contract

Each timed invocation includes reduction, the zero-mean basis, both spectral routes, and residual diagnostics. Mesh construction, assembly, constant-pressure interpolation and outer warmup/repeat checks are excluded. Wall time is reported. The calculation is rank-local on complete matrices, with no MPI collectives. These finite spectra do not prove mesh-uniform inf-sup stability.

Timing history

RodinMixedStabilityBenchmarks · real_time · slow · ranks: 1