Mathematics Seminar - 09/24/21

Abstract: This will be a two-part talk. In part 1, we present a C0 interior penalty finite element method for the sixth-order phase field crystal equation. We demonstrate that the numerical scheme is uniquely solvable, unconditionally energy stable, and convergent. We then present a few numerical experiments supporting these conclusions. In the second part of the talk, we present a numerical approximation method for the Cahn-Hilliard equation that incorporates continuous data assimilation in order to achieve long time accuracy. The method once again uses a C0 interior penalty spatial discretization of the fourth order Cahn-Hilliard equation, together with a semi-implicit temporal discretization. We prove the method is long time stable and long time accurate, for arbitrarily inaccurate initial conditions, provided enough data measurements are incorporated into the simulation. Numerical experiments illustrate the effectiveness of the method on a benchmark test problem.