Mathematical Problems in Engineering

Volume 2018, Article ID 7521273, 7 pages

https://doi.org/10.1155/2018/7521273

## A Cartesian Grid Method for Modeling Charge Distribution on Interfaces via Augmented Technique

^{1}School of Science, Nanjing University of Posts and Telecommunications, Nanjing, Jiangsu 210023, China^{2}Institute of Information Technology, Nanjing University of Chinese Medicine, Nanjing, Jiangsu 210023, China^{3}School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, China

Correspondence should be addressed to Qian Zhang; moc.361@52108891qz

Received 16 December 2017; Revised 1 April 2018; Accepted 8 April 2018; Published 14 May 2018

Academic Editor: Ruben Specogna

Copyright © 2018 Haifeng Ji 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.

#### Abstract

In the electrostatic field computations, second-order elliptic interface problems with nonhomogeneous interface jump conditions need to be solved. In realistic applications, often the total electric quantity on the interface is given. However, the charge distribution on the interface corresponding to the nonhomogeneous interface jump condition is unknown. This paper proposes a Cartesian grid method for solving the interface problem with the given total electric quantity on the interface. The proposed method employs both the immersed finite element with the nonhomogeneous interface jump condition and the augmented technique. Numerical experiments are presented to show the accuracy and efficiency of the proposed method.

#### 1. Introduction

We consider an isolated conductor that is placed near other charges or in an external electric field. For simplicity, we take a rectangular domain as the computational domain. The conductor occupying is included in the domain . The boundary of is denoted by which is also called an interface in this paper. We assume that . Let , let be the unit normal vector of pointing from to , and let be the permittivity of . The modeling of the electrostatic field leads to the interface problem [1],together with the interface jump conditions on the interface ,where the interfacial function represents the charge density on the interface and is also unknown; however, the total electric quantityis given. For simplicity, we consider a homogeneous Dirichlet boundary condition:Note that other boundary conditions can be treated using standard techniques. The function represents the external charge density of the part outside of the conductor. The solution is the potential. Once the potential is obtained, the electrostatic field can be computed by . If the charge distribution on the interface is given, then the potential can be solved efficiently by the immersed finite element (IFE) method for nonhomogeneous interface jump conditions (see [2, 3]). The significance of the IFE method [4–7] is the use of structured meshes which are independent of the interface, such as Cartesian meshes. The IFE method modifies the basis function on interface elements according to the interface conditions to capture the jumps of the exact solution. The weak form and the degrees of freedom remain the same as if there was no interface. If the coefficient is a constant without jumps, then the stiffness matrix is the same as that obtained by traditional finite element for the problem without interfaces. And only the right-hand side needs to be modified according to the interface conditions. The elliptic interface problem can be solved efficiently by the IFE method with the given nonhomogeneous interface jump .

However, for the problem discussed in this paper, only the total electric quantity on the interface is known, not the charge density distribution on the interface which is related to the nonhomogeneous interface jump condition of this problem. In [1], the authors proposed an iterative IFE method for this interface problem. An IFE method with nonhomogeneous interface jump conditions and a standard finite element method with ghost nodes are combined to get a “Prediction-Correction-Prediction” iteration. Numerical examples in [1] show that the iterative method is convergent and can solve this problem efficiently. Note that for partial differential equations there are many other methods in the literature [8–11]. In this paper, we present a new Cartesian grid method based on the IFE method and the augmented technique [12, 13]. By introducing the jump of the normal derivative of the exact solution as an augmented variable, we can get an efficient discretization in which the fast Fourier transform- (FFT-) based fast Poisson solver can be applied. The augmented variable is chosen such that the nonhomogeneous interface jump condition and the total electric quantity are satisfied. In the numerical method, the augmented variable is solved by using the GMRES iteration. Compared with the iterative IFE method proposed in [1], the advantage of our Cartesian grid method is that the FFT-based fast Poisson solver can be used. Numerical experiments are also provided in this paper to show the performance of the proposed method.

The rest of the paper is organized as follows. In Section 2, we describe the augmented technique for the interface problem with given electric quantity on the interface. We choose an augmented variable and rewrite the interface problem to a new one for which the FFT-based fast Poisson solver can be applied. In Section 3, we briefly recall the IFE method for the nonhomogeneous interface jump conditions, where the augmented variable is assumed to be given. In Section 4, the constraint of the total electric quantity is enforced and some implementation details about the GMRES iteration are described. Finally, some numerical examples are provided in Section 5 to show the accuracy and efficiency of the proposed method.

#### 2. Augmented Technique for Given Electric Quantity on Interfaces

By introducing an augmented variable and using the fact that the coefficient is a constant, the original problem can be written aswithIt is obvious that the solution of (5)-(6) is dependent on the augmented variable . Thus, we denote the solution of (5)-(6) by . Let the solution of the original model problem (1)-(4) be and define Then satisfies (5)-(6) with . In other words, . From the original model problem, the augmented variable should be chosen such that the solution of (5)-(6) satisfies This is the constraint for the choice of the augmented variable. When the conductor is in electrical equilibrium, the electrical potential is a constant and electric field inside the conductor; that is, on the interface . Hence, we have and .

In the continuous case, the augmented method is to find the solution of (5)-(6) and the augmented variable is constrained by

To present the numerical method, first we partition the domain into uniform rectangles with mesh size (, ); then we obtain the triangulation with mesh size by cutting the rectangles along one of diagonals in the same direction. We call an element an interface element if intersects ; otherwise, we call a noninterface element. The sets of all interface elements and noninterface elements are denoted by and , respectively. The interface is approximated by , the union of the line segments connecting the intersections of the interface and the edges of elements. That is,where is the intersection of the interface and edges of elements. Let be the domain with as its boundary, and . There is a small region around the interface , whose area is of order . And the distance between and satisfies .

In the discrete case, the augmented variable is piecewise constant and is defined at line segments on interface elements; that is, In other words, on the interface , the augmented variable is approximated by the piecewise constant function:

In the numerical method, if the vector (or in the function form) is given, then we can use the IFE method which will be described in the next section to get the discrete solution .

#### 3. Immersed Finite Element for Nonhomogeneous Interface Jump Conditions

In the IFE method, the interface jump conditions and are used to construct the discrete trial function space . For any , the finite dimensional function is piecewise linear on each element and is broken along to satisfy and on . For the test function space, we use the standard conforming finite element space . It will be shown later that the IFE function can be decomposed aswhere and is a piecewise linear function that is nonzero only on noninterface elements. Note that the function depends on the augmented variable .

Given the augmented variable , the IFE method for (5)-(6) is to find such thatThe discrete solution is . From (16), it is obvious that the stiffness matrix is the same as that obtained by traditional finite element for the problem without interfaces; only the right-hand side needs to be modified. Thus, we can take advantage of the fast Poisson solver to solve the system of equations efficiently when the augmented variable is known.

##### 3.1. Construction of the Function

First, we describe the function in the space in detail. On a non-interface element, is the standard linear function and the degrees of freedom are functional values on the vertices of the element. On interface elements, for example, on (see Figure 1 for an illustration), the function is constructed as the following piecewise linear function:The coefficients are chosen such that where the vector represents the degrees of freedom. Obviously, the function can be decomposed aswhere and satisfywith coefficients chosen such that