Fully implicit iterative solvers for free-surface Stokes problems — ice sheet dynamics, mantle convection, lava flow — diverge when using large time steps. The free surface moves, changing the domain geometry, which changes the Stokes solution, which moves the free surface further. The feedback loop amplifies errors faster than the solver can correct them.
The fix: add a stabilization term to the equations that vanishes upon convergence.
This is not an approximation. It's a mathematical scaffolding — a term that holds the iteration stable while the solver converges, then contributes exactly zero error to the final answer. The ghost term is present during construction and absent from the finished building. Only two Stokes solves per time step achieve second-order convergence, compared to the dozens required by explicit methods.
The authors prove this formally: the stabilization term is proportional to the difference between successive iterates. When the iterates converge, the difference goes to zero, and the term disappears. The solver converges because of something that isn't there in the final answer.
For ice sheet modelers, this means centennial-scale simulations can use time steps orders of magnitude larger than before without sacrificing accuracy. For the mathematics, it demonstrates something about the relationship between process and result: the method of arriving at the answer can contain elements that are not part of the answer itself, and removing those elements before convergence destroys the ability to arrive at the answer at all.
The through-claim: sometimes the thing that makes a solution possible is not part of the solution. Scaffolding that vanishes upon completion is not a trick — it's a structural necessity. The question is whether you recognize the ghost term for what it is: essential during the process, absent from the product, and not a contradiction.