The extend finite element method (XFEM) is popular in structural mechanics when dealing with the problem of the cracked domains. XFEM ends up with a linear system. However, XFEM usually leads to nonsymmetric and illconditioned stiff matrix. In this paper, we take the linear elastostatics governing equations as the model problem.
We propose a new iterative method to solve the linear equations. Here we separate two variables
Many physical phenomena can be modeled by partial differential equations with singularities and interfaces. Much attention has been focused on the development of interface problem in fluid dynamics and material science. The standard finite difference and finite element methods may not be successful in giving satisfactory numerical results for such problems. Hence, many new methods have been developed. Some of them are developed with the modifications in the standard methods, so that they can deal with the discontinuities and the singularities. Peskin developed immersed boundary method (IBM) [
In this paper, according to the characteristics of the stiffness matrix, we use an iterative method to avoid dealing with complicated stiff matrix. First we can use the result of linear equation from standard FEM. The result can be used to be the initial value for iteration. Then we decompose the unknown values to two parts; each part is governed by two linear equations, whose dimension is smaller than the whole stiff matrix. The new programm can be easily applied. Finally, numerical examples show that the proposed method is more efficient than common methods. Furthermore, the new method works well for some problems while the common method (such as LU) fails. The new iterative method can be applied to many other problems.
The outline of this paper is as follows. Section
The basic mathematical foundation of the partition of unity finite element method (PUFEM) was discussed in [
The first effort for developing the extended finite element method can be traced back to 1999, [
In mathematics, relevant theoretical parts of doing are almost same in [
Consider a domain
The total ordering of the element basis is given by
For example, we consider the linear elastostatics governing equations
For the finite element discretization, let
Consider the BubnovGalerkin implementation for the XFEM in twodimensional linear elasticity. In the XFEM, finitedimensional subspaces
There are several kinds of enrichment functions, such as absenrichment function, stepenrichment function, and Rampenrichment function. The choice of function is based on the behavior of the solution near the interface.
The Ramp function is defined as
The stepenrichment function is defined as
On substituting the trial and test functions form (
We can get the integral terms, and the matrices computed from the integral terms are block matrices of the form
The next step is solving the linear equations system.
We need to solve the following equations:
We must point out that the matrix
(i) We use the ordinary method to solve
(ii) Solve the small scale equation
The matrix
(iii) Then we use (
(iv) If the norm
In this part, we can use the common FEM solver such as LU or other useful solvers. We do not change the stiffness matrix
If the matrix
The iteration has the following advantages.
Other than the half bandwidth of zero elements do not have storage, nor involved in the calculation, the amount of storage crunch, the impact of rounding error is relatively small, even if it is illconditioned matrix, we often obtain a higher accuracy.
The operator only uses FEM linear equation solver; it is a simple calculation.
The original matrix deposit triangular matrix and diagonal matrix deposit area can overlap, thus avoiding to take up a lot of the middle unit of work, saving memory.
This solution can be extended to block direct solution.
With the iterative method, it is easy to build and test the general program and you can more accurately estimate the computation time; the user is relatively easy to grasp the algorithm process.
Also if we using in [
In this section, we report two experiments to verify the efficacy and accuracy of the new iterative method in XFEM. We use Matlab and modify software [
SFEM means the standard finite element method;
XFEM means the extended finite element method;
GMRES means the generalized minimal residual iterative method;
iterations mean the iterative times in the algorithm;
orders mean the numerical error order by XFEM:
Here
The exact solution is
Here
(a) The comparison of time by iteration and LU with different mesh. (b) The comparison of
(a) All nodes and enrichment nodes (in red color);
For the twodimensional solid test, we use the same enrichment function and shape function to compute (
We denote
Plate with a circular hole. Figure
DOF, sparsity of matrix, and condition number of linear equation by FEM and XFEM for Example

DOF_{FEM}  DOF_{XFEM}  conds_{FEM}  conds_{XFEM}  Sparsity of matrix (%) 


882  1018 


1.72 

3362  3626 


0.4774 

12800  13120 


0.222 

12800  13120 


0.13 

25600  26240 


0.0829 
The CPU time for different solvers in Example

FEM_{LU}  XFEM_{iteration}  XFEM_{LU} (s)  XFEM_{GMRES} (s) 


























The comparison of

FEM_{LU}  XFEM_{LU} (s)  XFEM_{GMRES}  XFEM_{iteration} (s) 



























Error_{XFEM}  Orders  CPU_{XFEM} (s)  Iterations 




1.313  24 


1.8992  6.52  19 


1.9489  20.61  19 


2.1541  59.07  19 


1.9028  145.92  18 
In this paper, we discuss how to use XFEM for the stiff matrix illconditioned and came to the following conclusions. Firstly, we found why some of problems cannot be computed by XFEM; the stiff matrix is not then symmetric and illconditioned. So, a lot of storage space is wasted. The time of computing XFEM equation is longer, which cannot be computed in certain subdivision. Secondly, we use XFEM for solving twodimensional solid test; when solving the XFEM equations, the iterative method and the standard finite element equations solver are adopted, so that we change the problems into solving the smaller scale equations and standard finite element stiffness matrix need not be changed. Finally, numerical examples show that the proposed method is more efficient than the common method. We can use this programm in many other interface problems later. Besides, we also consider combining the stable GFEM with our iterative method in the future. And it is believed that the XFEM can be widely used in more problems in fluid dynamics and material science. The calculation module in some software can be improved for users.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The author also thanks the anonymous reviewers for their comments in improving the paper. This research was supported by the special funds for NSFC (1127313, 11171269, 11201369, and 61163027) and the Ph.D. Programs Foundation of the Ministry of Education of China (Grant no. 20110201110027), and it was partially subsidized by the Fundamental Research Funds for the Central Universities (Grant nos. 08142003 and 08143007).