On the accuracy of finite element approximations to a class of interface problems

The jump is defined as [∇u · n] = ∇u · n +∇u ·n where u = u|Ω± and n is the unit outward pointing normal to Ω (see figure 1). Also, we denote [u] = u − u. Many numerical methods have been developed for problem (1.1). Perhaps the most notable ones are the finite difference method of Peskin [18] (i.e., immersed boundary method) and the method of LeVeque and Li [11] (i.e., the immersed interface method ; see also the method of Mayo [14, 15, 16]) .The method of LeVeque and Li [11] was developed for the more general problem with discontinuous diffusion coefficients, while the method of Peskin [18] was developed for fluid flow problems with an immersed boundary. Although the method of Peskin [18] is formulated with a force function F that incorporates the elastic force of the immersed boundary Γ, it was shown in [19] that it can be re-formulated as an interface problem (with α = 0) where β encodes the elastic force. Since the two important papers [18, 11] there have been many articles extending or improving these methods. In particular, finite element versions of these methods have appeared; see for example [3, 9, 6, 2]. For the above problem (α = 0), it is well known that the method of Peskin [18] is only first-order accurate whereas the method of LeVeque and Li [11] is second-order