The fictitious domain method with penalty for the parabolic problem in moving-boundary domain: The error estimate of penalty and the finite element approximation

We consider the fictitious domain method with penalty for the parabolic problem in a moving-boundary domain. Two types of penalty (the H1 and L2-penalty methods) are investigated, for which we obtain the error estimate of penalty. Moreover, for H1-penalty method, the H2-regularity and a-priori estimate depending on the penalty parameter ε are obtained. We apply the finite element method to the H1-penalty problem, and obtain the stability and error estimate for the numerical solution. The theoretical results are confirmed by the numerical experiments.