Improved Error Estimation for the Partially Penalized Immersed Finite Element Methods for Elliptic Interface Problems

Abstract

This paper is for proving that the partially penalized immersed finite element (PPIFE) methods developed in [25] converge optimally under the standard piecewise $H$² regularity assumption for the exact solution. In energy norms, the error estimates given in this paper are better than those in [25] where a stronger piecewise $H$³ regularity was assumed. Furthermore, with the standard piecewise $H$² regularity assumption, this paper proves that these PPIFE methods also converge optimally in the $L$² norm which could not be proved in [25] because of the excessive $H$³ regularity requirement.