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We propose a numerical approximation method for the Cahn-Hilliard equationsthat incorporates continuous data assimilation in order to achieve long timeaccuracy. The method uses a C$^0$ interior penalty spatial discretization ofthe fourth order Cahn-Hilliard equations, together with a backward Eulertemporal discretization. We prove the method is long time stable and long timeaccurate, for arbitrarily inaccurate initial conditions, provided enough datameasurements are incorporated into the simulation. Numerical experimentsillustrate the effectiveness of the method on a benchmark test problem.

Continuous data assimilation and long-time accuracy in a $C^0$ interior penalty method for the Cahn-Hilliard equation