Advances in Mathematical Physics
Volume 2018 (2018), Article ID 9891281, 9 pages
https://doi.org/10.1155/2018/9891281

Research Article

Enriched -Conforming Methods for Elliptic Interface Problems with Implicit Jump Conditions

Gwanghyun Jo and Do Y. Kwak

Department of Mathematical Sciences, KAIST, Daejeon 305-701, Republic of Korea

Correspondence should be addressed to Do Y. Kwak; kdy@kaist.ac.kr

Received 22 December 2017; Revised 13 March 2018; Accepted 28 March 2018; Published 9 May 2018

Academic Editor: Guozhen Lu

Copyright © 2018 Gwanghyun Jo and Do Y. Kwak. 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.

Abstract.

We develop a numerical method for elliptic interface problems with implicit jumps. To handle the discontinuity, we enrich usual -conforming finite element space by adding extra degrees of freedom on one side of the interface. Next, we define a new bilinear form, which incorporates the implicit jump conditions. We show that the bilinear form is coercive and bounded if the penalty term is sufficiently large. We prove the optimal error estimates in both energy-like norm and -norm. We provide numerical experiments. We observe that our scheme converges with optimal rates, which coincides with our error analysis.

1. Introduction

Interface problems arise in various disciplines including mechanical, material, and medical image and petroleum engineering [1–9]. There are several difficulties to solve for the governing equation of such problems.

Firstly, partial differential equations may have different coefficients along the interface due to the change of material properties. When the geometry of interface is complex, one needs to generate grids that align with the interfaces. Once a fitted grid is generated, one uses finite element method (FEM) or finite volume method (FVM) based on this grid. Secondly, the problem may have nonhomogeneous jump conditions along the interface. When the jumps along the interface are known explicitly, (say , with known and ), these jumps may be handled effectively by discontinuous Galerkin (DG) [10, 11] by incorporating jumps into the bilinear form with proper penalty terms. For example, an effective DG scheme was developed to describe discontinuous phenomena arising from porous media with discontinuous capillary pressure [12]. The interface problems with known jumps can be solved with immersed interface methods [13–15] or discontinuous bubble-immersed finite element methods [16].

However, when the jumps are implicit along the interface problems, numerically solving the governing equations becomes more challenging. Let us consider some problems with interface conditions, where the jumps of primary variables are related to the normal fluxes. Firstly, these problems arise in the medical imaging of cancer cells using MREIT [3, 4] or electrochemotherapy [17], where the jumps of an electric voltage across the cell membrane appear. Next, an elastic body has spring-type jumps that are related to stress [18, 19]. The heat in the material interface may have implicit jump conditions along the interface [20, 21]. Also, a generalized jump condition for Laplace equation or Helmholtz equation has been considered in [22–24].

The first attempt to solve the elliptic interface problems having implicit jump conditions seems to be introduced in [25], where the iterative method was used. Recently, some XFEM-based nonfitted methods were developed in [26, 27] for the elliptic problems and elasticity problems, respectively, where the extra degrees of freedom are introduced on elements cut by the interface. On the other hand, an immersed finite element type method was developed in [28].

In this work, we introduce a new numerical method to solve elliptic interface problems, where the jumps are related to the normal fluxes and some known functions. A main idea of our work is to include the jump conditions implicitly on the bilinear form so that the numerical solutions for the weak problems satisfy the implicit jump conditions. We enrich the usual FEM space near the interface. We show that our bilinear form is coercive and bounded and prove the optimal error estimates. In numerical section, we provide several numerical examples supporting our analysis.

Let be a convex domain in (), which is divided into and by a closed interface . The governing equations on are given by

where and , , , and is a positive piecewise constant; that is, in and in , where and are some positive constants. Here, is the outer unit normal vector to () and is the jump along the interface; that is, . Also, we define to be an outer normal vector to . The jump of normal derivatives of is defined as

We assume that is a positive constant.

We introduce some notations. Let be any domain and let , , be a usual Sobolev space with norm . We define as the set of functions in with vanishing trace on . We define the subspaces of ,

equipped with the (semi)norms:

Finally, we define subspace of :

We state a theorem regarding the existence and regularity of the problem [29, 30].

Theorem 1. Problem (1)–(4) has a unique solution such that, for some constant ,

The rest of the paper is organized as follows. In Section 2, we derive the variational forms for the problems with implicit jump conditions. We introduce new numerical methods in Section 3 and in Section 4 we prove the error estimates. In Section 5, we give numerical results that support our analysis. The conclusion follows in Section 6.

2. Variational Form

In this section, we derive a variational formulation of the model problem. Without loss of generality, we may assume that . First, we multiply to (1) and apply integration by parts on each subdomain to get

By summation, we have

Using the jump conditions (3) and (4), we see the second terms become

We define a bilinear form and a functional on :

where denotes inner product on and denotes the inner product. By (12) and (13), we have the weak problem: find satisfying

for all .

Now let us show that the weak problem (15) is equivalent to (1)–(4). Suppose that satisfies (15). First, let be any function (or ). Then, we have

By integration by parts, we see that satisfies

Now, assume that in (15). By Green’s theorem and the fact that , the left side of (15) becomes

Comparing with the right side of (15), we have

Finally, assume that in (15). By integration by parts, the left hand side of (15) becomes

Comparing with the right side of (15), we see that satisfies

3. Numerical Methods

In this section, we develop a numerical method for (1)–(4). Our method is obtained by adding extra degrees of freedom to -conforming space on one side of the interface. For simplicity, we assume that . However, similar constructions are possible for the case of as well.

Let be a given regular triangulation of fitted with the interface. We let and be set of elements in which belong to and , respectively. We let be the usual -conforming space; that is, any function in is continuous and piecewise linear and is vanishing on the boundary. We use notation for the set of linear functions on .

We let be the set of all neighboring elements of interface in ; that is, belongs to if and only if and at least one node of is located on . We let be the space of functions in vanishing on nodes not lying at the interface. For example, suppose that has three nodes , , and , where and are located on . Then, a function in is linear on vanishing at . In this case, has dimension two. On the other hand, if have only one node located on , the dimension of is one. A function in is extended to as follows:

For example, suppose that there are seven elements aligning with interface (see Figure 1). Then function in has a support on grey region. Moreover, has vanishing values on outside nodes on . Thus, has seven degrees of freedom, that is, .

Figure 1: The support and the degrees of freedom of a function in .

We decompose as

where belongs to and belongs to . We approximate from and approximate from . Thus, is approximated in .

Now, we discretize (1)–(4). Let be edges of elements in and let us define subspaces of (see Figure 1).

(i) is the set of edges of whose two endpoints are located on .(ii) is the set of edges of whose one endpoint is located on and the other is located in the interior .(iii).

We note that . For all edges in , we fix a normal vector once and for all. We define jumps and averages across the edges:

where and are two neighboring elements of .

We multiply to (1) and use integration by parts to obtain the following:

where

Let us classify into three categories. Firstly, if , then using the similar method used in deriving (12), we have

Secondly, if , then by using the identity

and the fact that

we have

Finally, if , vanishes, since both and are continuous across . Thus, (25) becomes

Now we propose our method based on enriched -conforming space: find satisfying

for all , where

In (33), the parameter is positive and the parameter is 1, 0, or , which is motivated by NIPG, IIPG, and SIPG of DG scheme [11].

We show that our scheme is consistent.

Lemma 2. Suppose that is the solution of (1)–(4). Then, for all , the following holds:

In other words, we have

for all .

Proof. Since for all , we have

for all . By (31), we have the conclusion.

4. Error Estimates

We define energy-like norm on .

Let be the usual trace operator. Then we have the following theorem [31, 32].

Lemma 3 (trace theorem). There exists a constant such that

for all .

We define a local interpolation operator by

where ’s are nodes of . The operator is extended to by for each element . We note that if belongs to , is discontinuous across ; that is, . However, is continuous on each subdomain and . Then, by the standard interpolation theory, there exists a constant such that

From this, we can obtain interpolation estimate in norm.

Corollary 4. There exists a constant such that, for all ,

Proof. Since is continuous across , we have

It suffices to show that

for some constant . Let . By the trace inequality (41), we have

We have the following coercivity property.

Theorem 5. If we choose so that

then the following holds:

for all , where

Proof. If , then by definition of and norm, we have

Now, suppose that or . We have

We bound by Cauchy-Schwarz inequality, (39), and Young’s inequality.

for all . Thus, we have

Then (47) is obtained by taking .

By a similar technique, we can show that is bounded.

Theorem 6. There exists a constant such that following holds:

for all .

Finally, we prove the error estimate in the energy-like norm.

Theorem 7. Assume that satisfies (46). There exists a constant such that

Proof. By (36), we have

for all . By (55), (47), and (53) we have

If we take , then, by (42) and (9), we have

Next, we prove estimates using duality argument when .

Theorem 8. If satisfies (46) and , there exists a constant such that

Proof. We define an auxiliary problem. Let be solution of

where . We multiply to (59) and we use integration by parts to have

We use similar techniques in the classification of of (25) to derive

Combining with (63) and (64) and the fact that is continuous on each subdomain and , we have

By definition of and the fact that is symmetric and by (36), (54), (42), and (9), we have

Thus, we have the conclusion.

5. Numerical Results

In this section, we provide some numerical experiments of elliptic interface problems with implicit jump conditions. We consider circle- and ellipse-type interface shapes.

We let the domain and we let be a triangulation of by regular triangles, which aligns with interfaces. We set in bilinear form (33). We take in (33) as a multiple of . We report the number of elements, degrees of freedom, -errors, and -errors for , , in Tables 1 and 2. For both examples, we observe optimal error convergence, which supports our analysis in Section 4.

Table 1: and errors of Example 1.

Elements
DoF

Order

Order

70

264

1.949

1.368

1062

1.988

1.163

4171

1.998

1.023

16557

1.998

1.002

65521

2.001

1.000

260961

2.000

1.002

Table 2: and errors of Example 2.

Elements
DoF

Order

Order

110

382

1.133

0.456

1355

2.162

1.034

5005

1.826

0.900

18993

1.988

0.959

74225

1.987

0.989

292404

1.976

0.986

Example 1 (circular interface). The interface is given by and and are inside and outside of , respectively. The coefficients are , , and . The exact solution is

where . We remark that satisfies jump conditions (3) and (4), where and are given as

Table 1 shows the number of elements and and piecewise errors. Figure 2 shows the numerical solution. We observe that our scheme has optimal convergence in and piecewise -norms.

Figure 2: Numerical solution of Example 1.

Example 2 (elliptical shape interface). The interface is given by and and are inside and outside of , respectively. The coefficients are and . In the previous example, was constant. However, in this example, we set as a function of :

The exact solution is

where . We remark that and in (3) and (4) are given as

Table 2 shows the errors and Figure 3 shows the numerical solution. Again, we observe the optimal convergence.

Figure 3: Numerical solution of Example 2.

6. Conclusion

In this work, we introduce a numerical method for elliptic interface problems, where the jumps are related to the normal fluxes. We enrich usual space by extra degrees of freedom on one side of the interface. We define bilinear form that includes the jump conditions implicitly. We prove that the bilinear form is coercive and bounded. Using Cea’s Lemma, we prove the error estimates in energy-like norm. Next, we prove error estimate using the duality arguments. We provide numerical experiments that support our analysis.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by NRF (Contract no. 2017R1D1A1B03032765).

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