Analysis of the fictitious domain method with penalty for elliptic problems

作者：Zhou Guanyu^*; Saito Norikazu

来源：Japan Journal of Industrial and Applied Mathematics, 2014, 31(1): 57-85.

DOI：10.1007/s13160-013-0124-2

• Fictitious domain methods
• Finite element methods
• Finite volume methods
• Error bounds

The fictitious domain method with H-1-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 H-1-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 H-1-penalty problems. The P1 finite element approximation to H-1-penalty problems is investigated. We study error estimates between the solutions of H-1-penalty problems and discrete problems in H-1 norm, as well as in L-2 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.

• 出版日期2014