  We consider a unique continuation problem where the Dirichlet trace of the
solution is known to have finite dimension. We prove Lipschitz stability of the
unique continuation problem and design a finite element method that exploits
the finite dimensionality to enhance stability. Optimal a priori and a
posteriori error estimates are shown for the method. The extension to problems
where the trace is not in a finite dimensional space, but can be approximated
to high accuracy using finite dimensional functions is discussed. Finally, the
theory is illustrated in some numerical examples.
