MPEMathematical Problems in Engineering1563-51471024-123XHindawi10.1155/2020/20819482081948Research ArticleOptimal Order Error Estimates of a Modified Nonconforming Rotated Q1 IFEM for Interface ProblemsYinPei1YueHongyun2https://orcid.org/0000-0002-0160-7889GuanHongbo3De LucaAlessandro1Business SchoolUniversity of Shanghai for Science and TechnologyShanghai 200093Chinausst.edu.cn2College of ScienceHenan University of TechnologyZhengzhou 450001Chinahaut.edu.cn3College of Mathematics and Information ScienceZhengzhou University of Light IndustryZhengzhou 450002Chinazzuli.edu.cn202023920202020200220202107202023920202020Copyright © 2020 Pei Yin et al.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper presents a new numerical method and analysis for solving second-order elliptic interface problems. The method uses a modified nonconforming rotated Q1 immersed finite element (IFE) space to discretize the state equation required in the variational discretization approach. Optimal order error estimates are derived in L2-norm and broken energy norm. Numerical examples are provided to confirm the theoretical results.

National Natural Science Foundation of China115015271130139271601119
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  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  obtained the error estimates of the infinite element method for the elliptic interface problems. However, the method proposed in  can only deal with the case that interfaces consist of straight lines. Chen and Zou  proved the error estimates in the energy norm and L2-norm of interface problems with the interface being C2-smooth. Moreover, it was shown that the error estimate in the energy norm could be optimal when the exact solution is much smoother (in W1,) near the interface (Remark 2.4 in ). Based on this assumption, the authors of  achieved the optimal energy norm and suboptimal L2-norm error estimates. Later on, the above result was applied to semilinear elliptic and parabolic interface problems in , and the optimal order error estimates of the energy norm were derived. And Guan and Shi  obtained the same convergence order by using the P1-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  modified an immersed centered FDM for elliptic interface problems defined on a simple domain and obtained the second-order 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.  constructed a class of quadratic IFE spaces and discussed the approximation capabilities for solving the second-order elliptic interface problems. The IFEM was applied in  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  for solving second-order 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  to be convergent to the exact solution, unfortunately, the error estimate in the energy norm was of order Oh1/2, which was not optimal, and the L2-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  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–Raviart-type 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 . Recently, some research studies showed the advantages of the NFEM for PDEs, such as . In , the authors introduced a nonconforming Crouzeix–Raviart IFEM to solve second-order 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  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 Q1 (cf. ) 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 Q1-element proposed in  cannot be used for this IFEM, although this element has some advantages over other elements for anisotropic noninterior meshes (cf. ). We should point out that the convergence order in the energy norm is half order higher than that in  and in which the L2-norm was not mentioned. Secondly, optimal order error estimates will be carried out in L2-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 Ω be a convex polygonal domain in R2, ΩΩ be an open domain with smooth curve bound ΓΩ, and Ω+=ΩΩ (see Figure 1). We consider the following elliptic interface problem:(1)βu=f,x=x1,x2Ω,uΩ=0,with the jump conditions at the interface Γ:(2)uΓ=0,βunΓ=0.

Sketch of the domain Ω=ΩΩ+.

In (2), uΓ denotes the jump of u across the interface Γ and n the outward unit normal vector to Γ. The coefficient β is a positive piecewise constant function defined by(3)βx=βs,xΩs,where s= or + throughout this paper.

The corresponding variational form of (1) is as follows. Find uH1Ω such that(4)au,v=f,v,vH01Ω,uΩ=0,which also satisfies the jump condition (2). In (4), au,v=Ωβuvdx,f,v=Ωfvdx.

Because of the low global regularity of the exact solution in (1), the following spaces and norms are defined as(5)PWm,pΩ=uΩsWm,pΩs,p1,m=0,1,2,equipped with the norm m,p,Ω=sum,p,Ωs2 and seminorm m,p,Ω=sum,p,Ωs2.

As the usual Sobolev spaces, when p=2, let(6)PHint2Ω=uΩsH2Ωs,βunΓ=0,equipped with the norm 2,Ω=su2,Ωs2 and seminorm 2,Ω=su2,Ωs2.

3. The Nonconforming IFE Space

In this section, local nonconforming Q1rot IFE basis functions will be introduced, and the well-posedness of the nonconforming IFE interpolation will be proven.

Assume that K^ is the square reference element with four vertices A^1=1,1,A^2=1,1,A^3=1,1, and A^4=1,1; four edges are F^1=A^1A^2¯,F^2=A^2A^3¯,F^3=A^3A^4¯, and F^4=A^4A^1¯. Define the FE K^,P^,Σ^ on K^ as follows:(7)P^=span1,x^1,x^2,x^12,Σ^=v^i,i=1,2,3,4,in which v^i=1/F^iF^iv^ds^.

Remark 1.

It is well known that the standard Q1 reference element K^,P^,Σ^ can be defined as(8)P^=span1,x^1,x^2,x^1x^2,Σ^=v^i,i=1,2,3,4,where v^i=vAi are the function values of v^x^ at the four vertices Ai of K^. The IFEM and the convergence analysis of this conforming bilinear element could be found in . This paper focuses on the IFEM of the nonconforming Q1rot element (7).

So, by direct calculation, the corresponding finite element interpolation function can be expressed as(9)Π^v^=3v^1v^2+3v^3v^44+v^2v^42x^1+v^3v^12x^2+3v^1+v^2v^3+v^44x^12.

Let φK:K^K be an invertible mapping from the reference element K^ to the general quadrilateral element K, and the FE space be defined as(10)Vh=vhv^h=vhK°φKP^,KTh,FvhFds=0,FK,where vhF=vh, while FΩ is the boundary edge. Let Πh:H1ΩVh be the associated interpolation operator on Vh; ΠK=ΠhK satisfies(11)FivΠKvds=0,i=1,2,3,4.

There holds the following interpolation error estimate for any given uH2Ω:(12)uΠhu0+huΠhuhch2u2,where vh1,h=KThvh1,K2 is a broken energy norm on Vh.

Consider the discrete variational form of (7) as follows. Find uhVh such that(13)ahuh,vh=f,vh,vhVh,where ahuh,vh=KThKβuhvhdx.

Now, some reasonable restrictions on the quadrilateral subdivision Th (0<h<1 be the mesh size) are given as follows. The interface is allowed to cut through the elements, which are named interface elements Kint. Otherwise, the elements are called noninterface elements Knon. For the interface elements Kint, assume that

The edges meet the interface at no more than two points

Each edge is passed through at most once except passed at two vertices

It is easy to check that Vh=VhintVhnon, where Vhint and Vhnon denote the FE spaces defined on interface elements Kint and noninterface elements Knon, respectively. In fact, the main concern is the interface elements Kint separated by the interface Γ into two subsets K and K+. The corresponding piecewise interpolation function should be constructed on K and K+, 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 Kint, we assume that the vertices are Ai, i=1,2,3,4. Without loss of generality, we assume that Kint intersects with Γ at two points D and E. 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 Vh has four freedoms (coefficients). The values vii=1,2,3,4 on K 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 D, E, and D+E/2. Intuitively, these eight conditions can yield the desired piecewise bilinear polynomial in an interface rectangle. In fact, since DE¯ can be considered as an approximation of the C2-curve DE˜, the interface is perturbed by a Oh2 term. From , one can see for the interpolation polynomial defined below, such a perturbation will only affect the interpolation error to the order of Oh2. This idea leads us to consider functions defined as follows:(14)Ψx=Ψx=ax1+bx2+c+dx12,xK,Ψ+x=a+x1+b+x2+c++d+x12,xK+,ΨD=Ψ+D,ΨE=Ψ+E,ΨD+E2=Ψ+D+E2,DE¯βΨnβ+Ψ+nds=0.

Lemma 1.

Given a reference interface element, the piecewise function defined by (14) is uniquely determined.

Proof.

For any function Ψ defined on a rectangular element K, we let Ψ^ be the corresponding function on K^ induced by Ψ with Ψ^x^=ΨφKx^.

Under this affine mapping φK, points D and E are mapped to(15)D^=1,y^0,E^=x^0,1,for Type I interface elements and(16)D^=1,y^0,E^=1^,y^1,for Type II interface elements.

So, we only need to prove that the desired result holds on the reference element.

The values Ψ^ii=1,2,3,4 on K^ provide four restrictions as follows:(17)Ψ^1=A^1E^ax1+bx2+c+dx12dx^1+E^A^2a+x1+b+x2+c++d+x12dx^2.

Remark 2.

If we choose P^=span1,x^1,x^2,x^22, the IFE space is also well-posed. However, the original rotated Q1 element proposed in  was adapted, P^ = span1,x^1,x^2,x^12x^22 instead of span1,x^1,x^2,x^12, and it can be checked that detBl=detBr=0, which will bring about the nonuniqueness of α^. Thus, this IFE space is not well-posed.

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 K be a general element with four edges Fii=1,2,3,4; then, for all uPHint2K,vhVh, we have(18)vhP0ivh0,Fich1/2vh1,K,βunP0iβun0,Fich1/2u2,K,where P0ivh=1/FiFivhds, here and later, and c is a generic positive constant independent of h.

Proof.

If K is a noninterface element, the results can be found in . Now, only the case of K is an interface element needs to be proven. Without loss of generality, we prove (18) for i=1.

Firstly, by the trace theorem, it can be derived that(19)vhP01vh0,F12chvh1,K2.

Secondly, let βu/n be the restriction of βu/n on K. Because of βu/nΓ=0, the function βu/n can be extended onto the whole element K, and a function βu˜/n can be obtained such that βu˜/n=βu/n in K (see  for the details). There holds(20)βun˜1,Kcβun1,K.

Thus, it can be derived that(21)βunP01βun0,F12=βunP01βun0,F12,=F1βunP01βun2dx1=hF1βu^n^P01βu^n^2dx^1=hβu^n^P^01βu^n^0,F^12chβu^n^P^01βu^n^0,K^2chβu^n^˜P^01βu^n^˜0,K^2chβu^n^˜1,K^2chβun˜1,K2chβun1,K2chu2,K2,where in the last fourth inequality, the norm equivalent property was used on a reference element, see Section 4 in  for details. The proof is completed.

Lemma 3.

Let uPHint2ΩH01Ω; then, for vhVh, there holds(22)KThKβunvhdschu2,Ωvh1,h.

Proof.

Noticing that KvhP0ivhds=0, there yields(23)KThKβunvhds=KThi=14FiβunP0iβunvhP0ivhds.

Applying Lemma 1 and Cauchy inequality to the right-hand side of (23) leads to the desired result.

Theorem 1.

Let uPHint2ΩH01Ω and uhVh be the solutions of (18) and (27), respectively; then, there hold(24)uuh1,hchu2,Ω,(25)uuh0,Ωch2u2,Ω.

Proof.

By Strong’s second lemma, it can be obtained that(26)uuh1,hcinfvhVhuvh1,h+supvhVhahu,vhf,vhvh1,h=cinfvhVhuvh1,h+supvhVhKThKu/nvhdsvh1,h.

From now on, Πh denotes the associated interpolation operator on Vhint or Vhnon. So, by the interpolation theory and Lemma 3, two error terms on the right-hand side of (26) can be bounded as(27)infvhVhuvh1,huΠhu1,hchu2,Ω,(28)supvhVhKThKu/nvhdsvh1,hchu2,Ω,respectively, which give the result (24).

In order to prove (25), we introduce the following auxiliary problem:(29)find wPHint2Ω such thatβw=uuh,x=x1,x2ΩwΩ=0,,with the jump conditions(30)wΓ=0,βwnΓ=0.

From , it is known that (29) has a unique solution w satisfying w2,Ωcuu00,Ω. Thus,(31)uuh0,Ω2=uuh,uuh=ahuuh,w+KThKβwnuuhds,=ahuuh,wΠhw+ahuuh,Πhw+KThKβwnuuhds=ahuuh,wΠhw+KThKβunwΠhwds+KThKβwnuuhdscuuh1,hwΠhw1,h+chu2,ΩwΠw1,h+chw2,Ωuuh1,hch2u2,Ωw2,Ωch2u2,Ωuuh0,Ω,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 , 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 P^1=span1,x^1,x^2,x^12 and P^2=span1,x^1,x^2,x^22 and compute the FE solutions uh1 and uh2, respectively. Then, using uh=1/2uh1+uh2 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 Ω=0,2×0,1, Ω = 0,1×0,1, and Ω+ = 1,2×0,1; the interface Γ occurs at x1=1. The exact solution u can be expressed as(32)u=u=sinπx1sinπx2,in Ω,u+=sinkπx1sinπx2,in Ω+,where β+=1 and β=k (k is an odd number) just satisfy the jump conditions. Now, the errors are listed in Tables 1 and 2 for k=5 and k=7, respectively. Figure 2 reports the convergence rates of our nonconforming IFEM in L2 and broken energy norms, respectively.

Errors in L2-norm and broken energy norm with k=5.

m×nuuh0,ΩOrderuuh1,hOrder
4×20.1740010202070.372938117902
8×40.0487543981021.8354914910.1941349984410.941876033
16×80.0138340291681.8173109660.0994134408480.965547404
32×160.0036799177361.9104759830.0507460337260.970145853
64×320.0009624510921.9348883800.0252355569811.007837138
128×640.0002340733072.0397528080.0126107582121.000802910
256×1280.0000626821981.9008327440.0063186030680.996977475

Errors in L2-norm and broken energy norm with k=7.

m×nuuh0,ΩOrderuuh1,hOrder
4×21.0251519809211.013510946696
8×40.2680720652561.9351450130.5326211223720.928180123
16×80.0734837508651.8671237160.2758503264140.949223956
32×160.0194460715691.9179465320.1450817763730.927019373
64×320.0050665768951.9403954750.0730898945570.989122457
128×640.0013879109351.8680963660.0379314414580.946277755
256×1280.0003275925452.0829405590.0196984568190.945311584

Convergence rates in L2-norm (a) and broken energy norm (b) for Example 1.

Example 2.

In this example, the domain is chosen as Ω=0,1×0,1, and the interface Γ occurs at x1+x2=1. Ω is the triangle rounded by x1-axis, x2-axis, and Γ. Ω+=ΩΩ. If the corresponding right-hand term is given as(33)f=f=4πx1cos2πx1+x24πx2cos2πx1+x2+8π2x1x2sin2πx1+x2,in Ω,f+=4kπx1cos2kπx1+x24kπx2cos2kπx1+x2+8k2π2x11x21sin2kπx1+x2,in Ω+,where k is a positive constant independent of x1 and x2, the exact solution can be expressed as(34)u=u=x1x2sin2πx1+x2,in Ω,u+=x11x21sin2kπx1+x2,in Ω+.

Obviously, uΩ=0. If β/β+=k, the jump conditions uΓ=0 and βu/nΓ=0 are satisfied. In the following, the errors are listed in Tables 1 and 2 for k=5 and k=7, respectively.

Figure 3 reports the convergence rates of our nonconforming IFEM in L2 and broken energy norm, respectively.

From Tables 14 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.

Convergence rates in L2-norm (a) and broken energy norm (b) for Example 2.

Errors in L2-norm and broken energy norm with k=5.

m×nuuh0,ΩOrderuuh1,hOrder
4×40.3293465076410.178346555746
8×80.0993682584371.7287492670.0947485463400.912507640
16×160.0279427816291.8303094360.0482110213540.974740816
32×320.0076214931721.8743300670.0248008110140.958975695
64×640.0018610504982.0339564700.0120825782541.037458961
128×1280.0004910299361.9222343150.0058175012981.054456806
256×2560.0001174086622.0642721330.0029147052610.997049624

Errors in L2-norm and broken energy norm with k=7.

m×nuuh0,ΩOrderuuh1,hOrder
4×40.4787336034850.524504640793
8×80.1366437515241.8088035860.2743580196610.934895790
16×160.0584186791381.2259178460.1460312387720.909782726
32×320.0187471663241.6397571930.0756972363380.947964487
64×640.0041396779132.1790821170.0389018067390.960403469
128×1280.0011827047681.8074285370.0199733848280.961758318
256×2560.0003252980411.8622559470.0097138017021.039970902
6. Conclusions

This paper discusses a modified nonconforming rotated Q1 IFEM for second-order elliptic interface problems. Optimal order error estimates of L2-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 parabolic-type or hyperbolic-type interface problems by using a suitable full discretization scheme. However, the method cannot be applied to other very popular nonconforming FEs, such as EQ1rot element , Carey element , and Wilson element .

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).

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