Analysis of the fictitious domain method with penalty for elliptic problems

The fictitious domain method with H1-penalty for elliptic problems is considered. We propose a new way to derive the sharp error estimates between the solutions of original elliptic problems and their H1-penalty problems, which can be applied to parabolic problem with moving-boundary maintaing the sharpness of the error estimate. We also prove some regularity theorems for H1-penalty problems. The P1 finite element approximation to H1-penalty problems is investigated. We study error estimates between the solutions of H1-penalty problems and discrete problems in H1 norm, as well as in L2 norm, which is not currently found in the literature. Thanks to regularity theorems, we can simplify the analysis of error estimates. Due to the integration on a curved domain, the discrete problem is not suitable for computation directly. Hence an approximation of the discrete problem is necessary. We provide an approximation scheme for the discrete problem and derive its error estimates. The validity of theoretical results is confirmed by numerical examples.