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 Open Access
The stable extrinsic extended finite element method for second order elliptic equation with interfaces
 Jianping Zhao^{1, 2}Email author,
 Yanren Hou^{3},
 Lina Song^{4} and
 Dongwei Gui^{2}
https://doi.org/10.1186/s1366201505174
© Zhao et al. 2015
 Received: 28 November 2014
 Accepted: 25 May 2015
 Published: 15 July 2015
Abstract
In this paper, a stable extrinsic extended finite element method (SXFEM) is proposed to solve the second order elliptic equation with discontinuous coefficients and interfaces. SXFEM is designed by the stable enrichment function and stress intensity factors (SIF)type enrichment functions. It shows that the proposed SXFEM can get the optimal convergence order. Numerical experiments are presented to verify the feasibility of the new method for this type of problem and the superiority compared with the standard FEM and XFEM.
Keywords
 extended finite element method
 generalized finite element method
 discontinuous coefficients
 extrinsic
 interface
1 Introduction
Another strategy of improvement is to enrich a polynomial approximation space so that the nonsmooth solutions can be modeled independent of the mesh. For example, the immersed boundary method (IBM) [3] and the immersed interface method (IIM) [4] are developed based on finite difference, and they modify the standard centered difference approximation to maintain the second order accuracy [5–11] or to get higher order methods [12, 13], while the immersed finite element method (IFEM) developed in [14–18] is designed to cope with interface problem based on the finite element method (FEM). In this paper, we consider the complex interface problems such as interfaces intersecting with each other.
Meanwhile, a variety of modifications to the conventional FEM have been made within the framework of the partition of unity (PU). One typical example is the extended finite element method (XFEM). It was first realized by Belytschko and Black in [19] by enriching the nodes of the finite elements near the crack tips and along the crack surfaces with the asymptotic crack tip functions. Since then, such a method received wide publicity and quick progress [20–23]. During the same decades, the generalized finite element method (GFEM) based on the partition of unity method (PUM) [24–26] was widely used to solve various types of problems. All of these methods share the property that they add special enrichment functions to a standard approximation space.
Based on and inspired by the development of these methods in [26], we try to use XFEM for solving elliptic problems with interfaces. One of our goals is to make the condition number of the matrix for the discrete system comparable with FEM by extrinsic XFEM.
The rest of the paper is organized as follows. Section 2 introduces preliminary definition related to the XFEM and the weak form of (1). The main part of this paper is Section 3, in which the feasible stable XFEM and its error estimation are derived. The integration strategy for XFEM is discussed in Section 4, and some numerical experiments are presented in Section 5 to show the feasibility of the proposed algorithms. A final conclusion is drawn in Section 6.
2 Preliminary definitions
2.1 The weak form of the problem
2.2 The extrinsic XFEM
3 Stable extrinsic extended finite element method and error estimation
In this section we give the stable extended finite element algorithm step by step and give the estimation for L2error and energyerror.
3.1 The stable extrinsic XFEM
Let \(a(x,y)\) in (1) be a piecewise constant, we will consider two situations, namely \(a(x,y)=a_{1}\) if \((x,y)\in \Omega_{1}\) and \(a(x,y)=a_{2}\) if \((x,y)\in\Omega_{2}\), where the subdomains have the interface: \(\Omega_{1}\cup\Omega_{2}=\Omega\), \(\Omega_{1}\cap\Omega_{2}=\Gamma\).
Algorithm 3.1
(i) Suppose that Ω is a rectangular domain. Find the firsttype enriched nodes set \(I^{\mathrm{en}}_{1}\) and elements along interfaces by a level set function. The secondtype enriched nodes set \(I^{\mathrm{en}}_{2}\) is chosen by the impact area of intersection. Meanwhile we can easily find the two types of enriched elements.
(vi) Output the numerical result and error.
Remark 3.1
The element stiff matrix size varies with different types of elements. The common element far away from interfaces has four freedoms, while the element near intersection has 24 degrees of freedoms.
Remark 3.2
When computing the integration on the element containing intersection, we decompose the element into several triangles by the location of intersection.
The XFEM is a PUM, where
(1) the patches \(\omega_{i}^{h}\) are ‘FE stars’ relative to a finite element (FE) rectangulation of Ω;
(2) the piecewise linear FE hat function \(N_{i}\) associated with the vertices of FE rectangularity serves as the PU.
Supposing \(u\in H^{1}_{0}(\Omega)\) is the solution of (1), we use Q1element as the PU function.
(1) If \(v_{i}\in S_{1}\), \(v_{j}\in S_{1}\), \(B_{ij}=B(N_{i},N_{j})\), which is the basic part of XFEM, is the standard \(N\times N\) FE stiffness matrix.
(2) If \(v_{i}\in S_{1}\), \(v_{j}\in S_{2}\) or \(v_{i}\in S_{2}\), \(v_{j}\in S_{1}\), \(B_{ij}=0\) and \(B_{ji}=0\), because the \(S_{1}\) and \(S_{2}\) are orthogonal in the inner product \(B(\cdot,\cdot)\).
3.2 The analysis of stable XFEM
Theorem 3.1
Theorem 3.2
Proof
4 Modified numerical integration for XFEM
In standard FEM, we often use standard Gauss integration in all elements because the shape functions are smooth in the inner of the element. However, if the problem has interface, the smoothness could not be guaranteed in some elements cut by an interface. In XFEM [28], give the outline of integration strategy.
5 Numerical test

SFEM means the standard finite element method,

SXFEM means the stable extended finite element method,

DOF means the degrees of freedom,

\(\uu_{h}\_{0}\): the relative L_{2}error for \(u_{h}\) using SXFEM,

\(\mathrm{order}_{L_{2}}\) means the convergence rate in L_{2}error order,

\(\uu_{h}\_{E}\): the relative energyerror for \(u_{h}\) using SXFEM, we get the result by$$\begin{aligned} \uu_{h}\_{E}= \biggl(\int_{\Omega}a(x,y) \bigl(\nabla(uu_{h}) \bigr)^{2}\,dx \,dy \biggr)^{1/2} \Big/ \biggl(\int_{\Omega}a(x,y) (\nabla u)^{2}\,dx \,dy \biggr)^{1/2}, \end{aligned}$$

\(\mathrm{order}_{E}\) means the convergence rate in energynorm.
Here, \(\beta=0.1\), it is a difficult problem for computation by standard FEM. The exact solution is singular on the origin node and the interfaces Γ are xaxis and yaxis (fixed by the discontinuity of \(a(x,y)\)). \(\Gamma_{1}:{(x,y)xy=0,x\geq0}\), \(\Gamma_{2}:{(x,y)xy=0,y\geq0}\), \(\Gamma_{3}:{(x,y)xy=0,x\leq0}\) and \(\Gamma_{4}:{(x,y)xy=0,y\leq0}\) The origin node \((0,0)\) is the intersecting interfaces.
 1.
The node with influence area \(\omega(x_{i})\cap\Gamma_{i}=\emptyset\);
 2.
The node with influence area \(\omega(x_{i})\), singularity node O, \(O\in \omega(x_{i}) \);
 3.
Other elements. Not all the nodes in these elements need to be enriched(one, two or three nodes are enriched in some elements).
DOF, the energyerror and L2error using sXFEM for different step, impact radius \(\pmb{r=0.1}\)
2/ h  DOF  \(\boldsymbol {\uu_{h}\_{0}}\)  \(\mathbf{order}_{\boldsymbol {L2}}\)  \(\boldsymbol {\uu_{h}\_{E}}\)  \(\mathbf{order}_{\boldsymbol {E}}\) 

19  416  3.14410816 × 10^{−4}  ∖  2.88823845 × 10^{−2}  ∖ 
39  1648  1.31537498 × 10^{−4}  2.1536  1.36757789 × 10^{−2}  1.0785 
79  6608  3.14410816 × 10^{−5}  2.0647  6.66191594 × 10^{−3}  1.0376 
159  26432  8.09627168 × 10^{−6}  1.9573  3.28872649 × 10^{−3}  1.0184 
DOF, the energyerror and L2error using sXFEM with different impact radius r
2/ h  r  DOF  \(\boldsymbol {\uu_{h}\_{0}}\)  \(\boldsymbol {\uu_{h}\_{E}}\) 

51  0.05  2720  7.62956091 × 10^{−5}  1.03929952 × 10^{−2} 
51  0.1  2800  7.63216118 × 10^{−5}  1.03929961 × 10^{−2} 
51  0.2  3024  7.64345134 × 10^{−5}  1.03929957 × 10^{−2} 
51  0.3  3456  8.01596131 × 10^{−5}  1.03930097 × 10^{−2} 
Comparison of L2error \(\pmb{(\uu_{h}\_{0})}\) using standard FEM, stable XFEM and modified intrinsic XFEM
2/ h  FEM  XFEM  SXFEM 

19  1.29367331 × 10^{−2}  5.85229689 × 10^{−4}  5.97809622 × 10^{−4} 
39  6.11857595 × 10^{−3}  1.31518929 × 10^{−4}  1.55184343 × 10^{−4} 
79  2.97964019 × 10^{−3}  3.12260721 × 10^{−5}  3.14410816 × 10^{−5} 
159  1.47079316 × 10^{−3}  7.61075872 × 10^{−6}  8.09627168 × 10^{−6} 
319  7.30756777 × 10^{−4}  1.87886932 × 10^{−6}  2.47196080 × 10^{−6} 
6 Conclusions
In this article, we discussed the stable XFEM for the second order elliptic equation with discontinuous coefficients and derivative of solutions, and it comes to the following conclusions. Firstly we modified the local enrichment function space, and we analyzed how the global error can be dominated by the local error. It was different from the shift function only changed in vertices [31]. Secondly we described the stable XFEM step by step, we also gave the error estimation if we use Q1 element. The L2error is \(o(h^{2})\), and energyerror is \(o(h)\). We also got the optimal convergence same with SXFEM. Two types of XFEM are better than FEM, to adapt the FEM result in reference [1] a triangular element was used. There the absolute energyerror was considered as 0.05, while the DOF is about 3,475. Numerical relative energyerror in this paper was \(1.36757789\times10^{2} \), while DOF is only 1,648. We will extend our method in general area, and it can be used to different meshes and high order polynomials. At last we gave the numerical simulation for the standard benchmark example. Numerical results support our analysis, we get the optimal order for energyerror and L2error, respectively.
Declarations
Acknowledgements
The authors of this work are grateful to the journal editors and the anonymous reviewers for their comments and recommendations, which have greatly improved our manuscript and made it more suitable for readers of the journal. This work is subsidized by China Postdoctoral Science Foundation funded project (No. 2014M562487) and NSFC of China (Nos. 11461068, 11171269, 11401332, 61163027, 11362021).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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