Error mechanisms intrinsic to legacy computational fluid dynamics (CFD) effectively compromise physics of fluids fidelity prediction of current CFD codes. Rigorously derived cubically nonlinear calculus alterations to Navier–Stokes (NS) partial differential equation (PDE) systems annihilate these legacy CFD error mechanisms, 1) spatial discretization-induced algebraic instability, 2) space-time discretization-generated phase aliasing, 3) artificial/numerical stabilization schemes, 4) discrete algebra theorization, 5) absence of local/global error estimates and error quantification, 6) physics-based Reynolds stress tensor modeling, 7) incompressible mass conservation, pressure, multiply connected domains, 8) iterative linear algebra admitted convergence stall. Weak formulation continuous Galerkin finite element (FE) algorithms for compressible and incompressible NS and the time-averaged/space-filtering altered PDE systems quantitatively validate these rigorous theoretical alteration leading to, 1) fourth order accuracy in physical space, wavenumber space and time on any mesh, 2) derivation of local/global error estimates and asymptotic convergence rates for FE p = 1,2,3 trial space basis implementations in NS PDE intrinsic norms, 3) analytically closed space-filtered NS PDE prediction of laminar profile velocity incipient transition, separation, turbulent profile reattachment then relaminarization, and finally, 4) matrix differential calculus identification of weak form nonlinear contributions to the quadratic convergent Newton iteration algorithm Jacobian.