A MODIFIED $P_1$ -- IMMERSED FINITE ELEMENT METHOD

Abstract

In recent years, the immersed finite element methods (IFEM) introduced in [?], [?] to solve elliptic problems having an interface in the domain due to the discontinuity of coefficients are getting more attentions of researchers because of their simplicity and efficiency. Unlike the conventional finite element methods, the IFEM allows the interface cut through the interior of the element, yet after the basis functions are altered so that they satisfy the flux jump conditions, it seems to show a reasonable order of convergence. In this paper, we propose an improved version of the P1 based IFEM by adding the line integral of flux terms on each element. This technique resembles the discontinuous Galerkin (DG) method, however, our method has much less degrees of freedom than the DG methods since we use the same number of unknowns as the conventional P1 finite element method. We prove H and L error estimates which are optimal both in order and regularity. Numerical experiments were carried out for several examples, which show the robustness of our scheme. Department of Mathematical Sciences, KAIST,, 291 Daehak-ro, Yuseong-gu, Daejeon, Korea 305-701. E-mail address: kdy@kaist.ac.kr Department of Mathematical Sciences, KAIST,, 291 Daehak-ro, Yuseong-gu, Daejeon, Korea 305-701. E-mail address: cool295@kaist.ac.kr 2000 Mathematics Subject Classification. primary 65N30, secondary 74S05, 76S05 .