Abstract
Abstract 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.
Highlights
The principle of the fictitious domain method is to solve the problem in a larger domain containing the domain of interest with a very simple shape
There exist some ways to derive the sharp error estimates for elliptic problems, it seems none of them has been applied to parabolic problem such that the sharpness of the error estimates are maintained
Our motivation lies in the study of the penalty fictitious domain method which can be applied to these time-dependent moving-boundary problems maintaining the sharpness of the error boundary
Summary
The principle of the fictitious domain method is to solve the problem in a larger domain (the fictitious domain) containing the domain of interest with a very simple shape. As a primary step towards this final end, we examine some new methods of error analysis for elliptic problems that can be applied to parabolic problems in time-dependent domain with sharp error estimate (which has been presented in our another paper [14]). There exist many works on finite element error estimate for elliptic problem with discontinuous coefficient or boundary unfitted mesh, we notice the discontinuous coefficient of H 1-penalty problem is dependent on the parameter , such that methods on those works may not be so suitable for our problem Most of those are not easy to apply to parabolic problem. Due to insufficient prior reported works on this issue in the literature, we derive some error estimates of the scheme to make the numerical analysis of the fictitious domain method with H 1-penalty more complete.
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