Inviscid aerofoil

• The lift coefficient is close to the compressible thin-aerofoil estimate $2\pi\alpha/\sqrt{1-M^2}$; thickness adds a few percent.
• The exact drag of a subsonic inviscid flow is zero, so the computed drag is error. It has three sources: discretization error (which the adjoint-based estimate in the last column measures, relative to the next polynomial degree), the singular trailing edge (which enrichment in $p$ resolves poorly) and the far-field boundary at a finite distance (a modelling error no mesh refinement removes). Comparing the estimate with the drag itself shows how much of it better resolution would recover.
• The derivative columns come from three adjoint solves (one per output) and include $\partial/\partial\alpha$, $\partial/\partial M$ and the sensitivity to every surface node.

The residual falls slowly while the pseudo-time step grows, then at Newton’s rate — the Jacobian is exact — down to round-off. The p-multigrid hierarchy for $p = 3$ is $(3, 1, 0)$.

One adjoint solve returns $\mathrm dc_l/\mathrm d\mathbf x$ for every surface geometry node. The chart shows its vertical component along the chord: where pushing the surface up raises the lift, and by how much. A geometry parameterisation maps design variables to node displacements and contracts them with this field (MDO integration).

Lift responds strongly to changes of the effective camber line near the trailing edge (the Kutta condition) and to the nose shape. Individual node sensitivities also scale with the surface area a node represents, so their magnitude follows the node clustering; a smooth parameterisation integrates over this.