Optimal convergence analysis of an immersed interface finite element method

Abstract

We analyze an immersed interface finite element method based on linear polynomials on noninterface triangular elements and piecewise linear polynomials on interface triangular elements. The flux jump condition is weakly enforced on the smooth interface. Optimal error estimates are derived in the broken H 1 -norm and L 2 -norm.

A Modified $P_1$ -- Immersed Finite Element Method

“…However, more numerical • , β − = 10, β + = 1 experiments show the original P 1 -IFEM is not optimal for some problems. The proof in [10] seems incorrect. However, our modified scheme is always robust for all problems tested including unreported ones).…”

“…These are continuous, piecewise linear functions on T satisfying the flux jump condition along DE, whose uniqueness and existence are known [10], [20]. We denote by S h (T ) the space of functions generated byφ i , i = 1, 2, 3 constructed above.…”

“…These methods use meshes which do not necessarily align with the discontinuities of the coefficients [20], [21], thus violate a basic principle of triangulations in the conventional finite element methods [4], [11]. However, when the basis functions are modified so that they satisfy the interface conditions, they seem to work well [10], [20], [21]. These methods were extended to the case of Crouzeix-Raviart P 1 nonconforming finite element method [12] by Kwak et al [18], and to the problems with nonzero jumps in [7].…”

Some error estimates on the large jump asymptotic approach for the interface problem

“…If the interface is smooth enough, then the solution of the interface problem is also very smooth in individual regions where the coefficient is smooth, but due to the jump of the coefficient across the interface, the global regularity is usually low. Because of the low global regularity and the irregular geometry of the interface, achieving accuracy is difficult with standard finite element methods, unless the elements fit with the interface of general shape [1]. It is well known that some popular methods are put out to efficiently solve this type of interface problems, such as the immersed interface method [1][2][3], the average methods [4,5], the finite element methods [6], the finite difference method and the mixed finite element method [7,8], the asymptotic expansion method [9], and so on.…”

Accurate gradient computations at interfaces using finite element methods

“…The immersed finite element method (IFEM) was developed for 1D and 2D interface problems by Li (1998) as well as Li and Wu (2003), respectively. Since then, many IFEMs and related analysis have appeared in the literature (see, e.g., Chou and Wee, 2010;He and Lin, 2011;Ji and Li, 2016), along with applications (An and Chen, 2014;Lin and Zhang, 2012;2015, Yang, 2002. Often they provide a second order accurate solution in the L 2 norm but only a first order accurate flux.…”