Advances in Mathematical Physics

Volume 2018, Article ID 9891281, 9 pages

https://doi.org/10.1155/2018/9891281

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

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

Correspondence should be addressed to Do Y. Kwak; rk.ca.tsiak@ydk

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 bywhere 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 becomeWe 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 satisfyingfor 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, .