Research Article  Open Access
Pei Yin, Hongyun Yue, Hongbo Guan, "Optimal Order Error Estimates of a Modified Nonconforming Rotated IFEM for Interface Problems", Mathematical Problems in Engineering, vol. 2020, Article ID 2081948, 9 pages, 2020. https://doi.org/10.1155/2020/2081948
Optimal Order Error Estimates of a Modified Nonconforming Rotated IFEM for Interface Problems
Abstract
This paper presents a new numerical method and analysis for solving secondorder elliptic interface problems. The method uses a modified nonconforming rotated immersed finite element (IFE) space to discretize the state equation required in the variational discretization approach. Optimal order error estimates are derived in norm and broken energy norm. Numerical examples are provided to confirm the theoretical results.
1. Introduction
Interface problems arise in many applications, such as mechanical analysis in material sciences or fluid dynamics, where two distinct materials or fluids with different conductivities or densities encounter on an interface [1, 2]. Because of the discontinuity of the properties along the interface and different control equations corresponding to different materials, the solutions of such problems have low regularity on the whole physical domain. Hence, it is a challenge to develop efficient numerical methods for such elliptic interface problems.
In recent years, the convergence analysis of the finite element method (FEM) for the interface problems has been discussed in many publications. For instance, Babuška [3] investigated the elliptic interface problem with a smooth interface. By the use of the boundary and jump conditions incorporated in the cost functions, an equivalent minimization problem was built, and error estimates in the energy norm were derived. Han [4] obtained the error estimates of the infinite element method for the elliptic interface problems. However, the method proposed in [4] can only deal with the case that interfaces consist of straight lines. Chen and Zou [5] proved the error estimates in the energy norm and norm of interface problems with the interface being smooth. Moreover, it was shown that the error estimate in the energy norm could be optimal when the exact solution is much smoother (in ) near the interface (Remark 2.4 in [5]). Based on this assumption, the authors of [6] achieved the optimal energy norm and suboptimal norm error estimates. Later on, the above result was applied to semilinear elliptic and parabolic interface problems in [7], and the optimal order error estimates of the energy norm were derived. And Guan and Shi [8] obtained the same convergence order by using the nonconforming triangular element.
Immersed methods are effective for solving interface problems, which contain immersed finite difference method (IFDM) and immersed finite element method (IFEM). LeVeque and Li [9] modified an immersed centered FDM for elliptic interface problems defined on a simple domain and obtained the secondorder accuracy on the uniform grid. Moreover, this IFDM was applied to Stokes flow interface problems and moving interface problems, respectively (cf. [10, 11]). Because the FEM has many advantages in engineering calculations, such as the low requirements for the smoothness of the exact solutions, the flexibility of the divisions, and the generality of application programs, a lot of research studies have been devoted to the IFEM based on the basic idea of adopting a proper FE space to conquer the trouble caused by the interface. For example, Camp et al. [12] constructed a class of quadratic IFE spaces and discussed the approximation capabilities for solving the secondorder elliptic interface problems. The IFEM was applied in [13] to solve elliptic interface problems with nonhomogeneous jump conditions for the Galerkin formulation, which can be considered as an extension of those IFEMs in the literature developed for homogeneous jump conditions. A bilinear IFE space was proposed in [14] for solving secondorder elliptic boundary value problems, and the error estimates were given for the interpolation, which indicated that this IFE space has the usual approximation capability. Although the IFE solution was proved in [15] to be convergent to the exact solution, unfortunately, the error estimate in the energy norm was of order , which was not optimal, and the norm error estimate was not considered. The above linear and bilinear IFEM was also applied to elliptic interface optimal control problems, Stokes interface problems, and eigenvalue interface problems, see [16–18] for details. In addition, the above IFEMs are all denoted to conforming linear or bilinear FE approximations.
However, the works mentioned above are mainly contributions to the conforming FEM. In fact, nonconforming finite element method (NFEM) has some advantages compared with the conforming ones. For example, for the Crouzeix–Raviarttype nonconforming finite element, since the unknowns of the elements are associated with the element edges or faces, each degree of freedom belongs to at most two elements, so using the nonconforming elements facilitates the exchange of information across each subdomain and provides spectral radius estimates for the iterative domain decomposition operator, see [19]. Recently, some research studies showed the advantages of the NFEM for PDEs, such as [20–22]. In [23], the authors introduced a nonconforming Crouzeix–Raviart IFEM to solve secondorder elliptic problems, but the interface elements were still constructed by conforming the linear triangular element. In this paper, we will adapt the nonconforming finite element different from [23] to deal with the interface problems over all the domains considered.
The remainder of the paper is organized as follows. Firstly, a nonconforming modified rotated (cf. [24]) IFE space will be constructed for the elliptic interface problems, which just satisfies the jump conditions. And a remark is given to explain that the original rotated element proposed in [25] cannot be used for this IFEM, although this element has some advantages over other elements for anisotropic noninterior meshes (cf. [26]). We should point out that the convergence order in the energy norm is half order higher than that in [15] and in which the norm was not mentioned. Secondly, optimal order error estimates will be carried out in norm and broken energy norm by employing some novel analysis techniques. Finally, some numerical results are provided to verify our theoretical analysis.
2. Elliptic Interface Problem
Let be a convex polygonal domain in , be an open domain with smooth curve bound , and (see Figure 1). We consider the following elliptic interface problem:with the jump conditions at the interface :
In (2), denotes the jump of across the interface and the outward unit normal vector to . The coefficient is a positive piecewise constant function defined bywhere or throughout this paper.
The corresponding variational form of (1) is as follows. Find such thatwhich also satisfies the jump condition (2). In (4), .
Because of the low global regularity of the exact solution in (1), the following spaces and norms are defined asequipped with the norm and seminorm .
As the usual Sobolev spaces, when , letequipped with the norm and seminorm .
3. The Nonconforming IFE Space
In this section, local nonconforming IFE basis functions will be introduced, and the wellposedness of the nonconforming IFE interpolation will be proven.
Assume that is the square reference element with four vertices , and ; four edges are , and . Define the FE on as follows:in which .
Remark 1. It is well known that the standard reference element can be defined aswhere are the function values of at the four vertices of . The IFEM and the convergence analysis of this conforming bilinear element could be found in [13–15]. This paper focuses on the IFEM of the nonconforming element (7).
So, by direct calculation, the corresponding finite element interpolation function can be expressed asLet be an invertible mapping from the reference element to the general quadrilateral element , and the FE space be defined aswhere , while is the boundary edge. Let be the associated interpolation operator on ; satisfiesThere holds the following interpolation error estimate for any given :where is a broken energy norm on .
Consider the discrete variational form of (7) as follows. Find such thatwhere .
Now, some reasonable restrictions on the quadrilateral subdivision ( be the mesh size) are given as follows. The interface is allowed to cut through the elements, which are named interface elements . Otherwise, the elements are called noninterface elements . For the interface elements , assume that(1)The edges meet the interface at no more than two points(2)Each edge is passed through at most once except passed at two verticesIt is easy to check that , where and denote the FE spaces defined on interface elements and noninterface elements , respectively. In fact, the main concern is the interface elements separated by the interface into two subsets and . The corresponding piecewise interpolation function should be constructed on and , respectively. The key is how to make them together so that the jump conditions across the interface are maintained.
To describe the local IFE space on an interface element , we assume that the vertices are , . Without loss of generality, we assume that intersects with at two points and . There are two types of rectangle interface elements. Type I interface elements are those for which the interface intersects with two of their adjacent edges; Type II interface elements are those for which the interface intersects with two of their opposite edges.
Note that each piecewise polynomial in has four freedoms (coefficients). The values on provide four restrictions. The normal derivative jump condition on DE provides another restriction. Then, three more restrictions can be provided by requiring the continuity of the finite element function at interface points , , and . Intuitively, these eight conditions can yield the desired piecewise bilinear polynomial in an interface rectangle. In fact, since can be considered as an approximation of the curve , the interface is perturbed by a term. From [15], one can see for the interpolation polynomial defined below, such a perturbation will only affect the interpolation error to the order of . This idea leads us to consider functions defined as follows:
Lemma 1. Given a reference interface element, the piecewise function defined by (14) is uniquely determined.
Proof. For any function defined on a rectangular element K, we let be the corresponding function on induced by with .
Under this affine mapping , points and are mapped tofor Type I interface elements andfor Type II interface elements.
So, we only need to prove that the desired result holds on the reference element.
The values on provide four restrictions as follows:
Remark 2. If we choose , the IFE space is also wellposed. However, the original rotated element proposed in [25] was adapted, = instead of , and it can be checked that , which will bring about the nonuniqueness of . Thus, this IFE space is not wellposed.
4. Convergence Analysis for the Elliptic Interface Problem
In this section, the convergence analysis and error estimates of the IFEM will be carried out for the elliptic interface problem. In order to do this, the following two important lemmas are proven as follows.
Lemma 2. Let be a general element with four edges ; then, for all , we havewhere , here and later, and is a generic positive constant independent of .
Proof. If is a noninterface element, the results can be found in [27]. Now, only the case of is an interface element needs to be proven. Without loss of generality, we prove (18) for .
Firstly, by the trace theorem, it can be derived thatSecondly, let be the restriction of on . Because of , the function can be extended onto the whole element , and a function can be obtained such that in (see [14] for the details). There holdsThus, it can be derived thatwhere in the last fourth inequality, the norm equivalent property was used on a reference element, see Section 4 in [28] for details. The proof is completed.
Lemma 3. Let ; then, for , there holds
Proof. Noticing that , there yieldsApplying Lemma 1 and inequality to the righthand side of (23) leads to the desired result.
Theorem 1. Let and be the solutions of (18) and (27), respectively; then, there hold
Proof. By Strong’s second lemma, it can be obtained thatFrom now on, denotes the associated interpolation operator on or . So, by the interpolation theory and Lemma 3, two error terms on the righthand side of (26) can be bounded asrespectively, which give the result (24).
In order to prove (25), we introduce the following auxiliary problem:with the jump conditionsFrom [28], it is known that (29) has a unique solution satisfying . Thus,which leads to (25). The proof is completed.
Remark 3. The error estimate order in Theorem 1 is optimal and is half order higher than [15], which benefits from the proper partition near the interface.
Remark 4. In order to conquer the asymmetry of the basis function space, one can first choose and and compute the FE solutions and , respectively. Then, using as the approximation solution also satisfies Theorem 1.
5. Numerical Results
In this section, two numerical experiments will be carried out for an elliptic interface problem.
Example 1. (see [6, 29]). In this example, the domain is chosen as , = , and = ; the interface occurs at . The exact solution can be expressed aswhere and ( is an odd number) just satisfy the jump conditions. Now, the errors are listed in Tables 1 and 2 for and , respectively. Figure 2 reports the convergence rates of our nonconforming IFEM in and broken energy norms, respectively.


(a)
(b)
Example 2. In this example, the domain is chosen as , and the interface occurs at . is the triangle rounded by axis, axis, and . . If the corresponding righthand term is given aswhere is a positive constant independent of and , the exact solution can be expressed asObviously, . If , the jump conditions and are satisfied. In the following, the errors are listed in Tables 1 and 2 for and , respectively.
Figure 3 reports the convergence rates of our nonconforming IFEM in and broken energy norm, respectively.
From Tables 1–4 and Figures 2 and 3, one can see that the proposed new nonconforming IFEM can solve the linear interface problem with the optimal order error estimates, but how to apply this method to the nonlinear case still remains open.
(a)
(b)


6. Conclusions
This paper discusses a modified nonconforming rotated IFEM for secondorder elliptic interface problems. Optimal order error estimates of norm and broken energy norm are derived. Numerical examples are provided to confirm the theoretical results.
We should point out that this method is suitable for parabolictype or hyperbolictype interface problems by using a suitable full discretization scheme. However, the method cannot be applied to other very popular nonconforming FEs, such as element [22], element [30], and element [31].
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant nos. 11501527, 11301392, and 71601119).
References
 G. Hetzer and A. Meir, “On an interface problem with a nonlinear jump condition, numerical approximation of solutions,” International Journal of Numerical Analysis and Modeling, vol. 4, no. 34, pp. 519–530, 2007. View at: Google Scholar
 G. Hou, J. Wang, and A. Layton, “Numerical methods for fluidstructure interaction—a review,” Communications in Computational Physics, vol. 12, no. 2, pp. 337–377, 2012. View at: Publisher Site  Google Scholar
 I. Babuska, “The finite element method for elliptic equations with discontinuous coefficients,” Computing, vol. 5, no. 3, pp. 207–213, 1970. View at: Publisher Site  Google Scholar
 H. Han, “The numerical solutions of interface problems by infinite element method,” Numerische Mathematik, vol. 39, no. 1, pp. 39–50, 1982. View at: Publisher Site  Google Scholar
 Z. Chen and J. Zou, “Finite element methods and their convergence for elliptic and parabolic interface problems,” Numerische Mathematik, vol. 79, no. 2, pp. 175–202, 1998. View at: Publisher Site  Google Scholar
 R. K. Sinha and B. Deka, “A priori error estimates in the finite element method for nonselfadjoint elliptic and parabolic interface problems,” Calcolo, vol. 43, no. 4, pp. 253–277, 2006. View at: Publisher Site  Google Scholar
 R. K. Sinha and B. Deka, “Finite element methods for semilinear elliptic and parabolic interface problems,” Applied Numerical Mathematics, vol. 59, no. 8, pp. 1870–1883, 2009. View at: Publisher Site  Google Scholar
 H. Guan and D. Shi, “P_{1}nonconforming triangular finite element method for elliptic and parabolic interface problems,” Applied Mathematics and Mechanics, vol. 36, no. 9, pp. 1197–1212, 2015. View at: Publisher Site  Google Scholar
 R. J. LeVeque and Z. Li, “The immersed interface method for elliptic equations with discontinuous coefficients and singular sources,” SIAM Journal on Numerical Analysis, vol. 31, no. 4, pp. 1019–1044, 1994. View at: Publisher Site  Google Scholar
 R. J. LeVeque and Z. Li, “Immersed interface methods for Stokes flow with elastic boundaries or surface tension,” SIAM Journal on Scientific Computing, vol. 18, no. 3, pp. 709–735, 1997. View at: Publisher Site  Google Scholar
 Z. L. Li, “Immersed interface method for moving interface problems,” Numerical Algorithms, vol. 14, no. 4, pp. 269–293, 1997. View at: Google Scholar
 B. Camp, T. Lin, Y. Lin, and W. Sun, “Quadratic immersed finite element spaces and their approximation capabilities,” Advances in Computational Mathematics, vol. 24, no. 1–4, pp. 81–112, 2006. View at: Publisher Site  Google Scholar
 X. M. He, T. Lin, and Y. P. Lin, “Immersed finite element methods for elliptic interface problems with nonhomogeneous jump conditions,” International Journal of Numerical Analysis and Modeling, vol. 8, no. 2, pp. 284–301, 2010. View at: Google Scholar
 X. He, T. Lin, and Y. Lin, “Approximation capability of a bilinear immersed finite element space,” Numerical Methods for Partial Differential Equations, vol. 24, no. 5, pp. 1265–1300, 2008. View at: Publisher Site  Google Scholar
 X. He, T. Lin, and Y. Lin, “The Convergence of the bilinear and linear immersed finite element solutions to interface problems,” Numerical Methods for Partial Differential Equations, vol. 28, no. 1, pp. 312–330, 2012. View at: Publisher Site  Google Scholar
 Q. Zhang, K. Ito, Z. Li, and Z. Zhang, “Immersed finite elements for optimal control problems of elliptic PDEs with interfaces,” Journal of Computational Physics, vol. 298, no. 10, pp. 305–319, 2015. View at: Publisher Site  Google Scholar
 A. Slimane, C. Nabil, and T. Lin, “An immersed discontinuous finite element method for Stokes interface problems,” Computer Methods in Applied Mechanics and Engineering, vol. 293, no. 8, pp. 170–190, 2015. View at: Publisher Site  Google Scholar
 S. Lee, D. Y. Sim, and I. Simb, “Immersed finite element method for Eigenvalue problem,” Journal of Computational and Applied Mathematics, vol. 313, no. 3, pp. 410–426, 2017. View at: Publisher Site  Google Scholar
 J. Douglas Jr., J. E. Santos, D. Sheen, and X. Ye, “Nonconforming Galerkin methods based on quadrilateral elements for second order elliptic problems,” ESAIM: Mathematical Modelling and Numerical Analysis, vol. 33, no. 4, pp. 747–770, 1999. View at: Publisher Site  Google Scholar
 H. Guan and D. Shi, “A stable nonconforming mixed finite element scheme for elliptic optimal control problems,” Computers & Mathematics with Applications, vol. 70, no. 3, pp. 236–243, 2015. View at: Publisher Site  Google Scholar
 H. Guan, D. Shi, and X. Guan, “High accuracy analysis of nonconforming MFEM for constrained optimal control problems governed by Stokes equations,” Applied Mathematics Letters, vol. 53, no. 3, pp. 17–24, 2016. View at: Publisher Site  Google Scholar
 H. B. Guan and D. Y. Shi, “An efficient NFEM for optimal control problems governed by a bilinear state equation,” Computers and Mathematics with Applications, vol. 77, no. 1, pp. 1821–1827, 2019. View at: Publisher Site  Google Scholar
 S. Y. Wang and H. Z. Chen, “An immersed finite element method based on CrouzeixRaviart elements,” Mathematica Numerica Sinaca, vol. 34, no. 1, pp. 125–138, 2012. View at: Google Scholar
 T. Apel, S. Nicaise, and J. Schöberl, “CrouzeixRaviart type finite elements on anisotropic meshes,” Numerische Mathematik, vol. 89, no. 2, pp. 193–223, 2001. View at: Publisher Site  Google Scholar
 R. Rannacher and S. Turek, “Simple nonconforming quadrilateral Stokes element,” Numerical Methods for Partial Differential Equations, vol. 8, no. 2, pp. 97–111, 1992. View at: Publisher Site  Google Scholar
 S. Mao and Z. Shi, “Nonconforming rotated Q_{1} element on nontensor product anisotropic meshes,” Science in China Series A: Mathematics, vol. 49, no. 10, pp. 1363–1375, 2006. View at: Publisher Site  Google Scholar
 D. Y. Shi, S. P. Mao, and S. C. Chen, “An anisotropic nonconforming finite element with some superconvergence results,” Journal of Computational Mathematics, vol. 23, no. 3, pp. 261–274, 2005. View at: Google Scholar
 S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, SpringerVerlag, Berlin, Germany, 1994.
 C. Attanayake and D. Senaratne, “Convergence of an immersed finite element method for semilinear parabolic interface problems,” Applied Mathematical Sciences, vol. 5, no. 3, pp. 135–147, 2011. View at: Google Scholar
 G. F. Carey, “An analysis of finite element equations and mesh subdivision,” Computer Methods in Applied Mechanics and Engineering, vol. 9, no. 2, pp. 165–179, 1976. View at: Publisher Site  Google Scholar
 R. L. Taylor, P. J. Beresford, and E. L. Wilson, “A nonconforming element for stress analysis,” International Journal for Numerical Methods in Engineering, vol. 10, no. 6, pp. 1211–1219, 1976. View at: Publisher Site  Google Scholar
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