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

Finite element approximation of unique continuation of functions with finite dimensional trace