Abstract and Applied Analysis

Volume 2014 (2014), Article ID 780769, 11 pages

http://dx.doi.org/10.1155/2014/780769

Research Article

Numerical Solution for Elliptic Interface Problems Using Spectral Element Collocation Method

Peyman Hessari,¹ Sang Dong Kim,² and Byeong-Chun Shin³

¹Department of Mathematical Sciences, Ulsan National Institute of Science and Technology, Ulsan 689-798, Republic of Korea

²Department of Mathematics, Kyungpook National University, Daegu 702-701, Republic of Korea

³Department of Mathematics, Chonnam National University, Gwangju 500-757, Republic of Korea

Received 24 February 2014; Revised 26 May 2014; Accepted 2 June 2014; Published 24 June 2014

Academic Editor: Sofiya Ostrovska

Copyright © 2014 Peyman Hessari 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.

The aim of this paper is to solve an elliptic interface problem with a discontinuous coefficient and a singular source term by the spectral collocation method. First, we develop an algorithm for the elliptic interface problem defined in a rectangular domain with a line interface. By using the Gordon-Hall transformation, we generalize it to a domain with a curve boundary and a curve interface. The spectral element collocation method is then employed to complex geometries; that is, we decompose the domain into some nonoverlaping subdomains and the spectral collocation solution is sought in each subdomain. We give some numerical experiments to show efficiency of our algorithm and its spectral convergence.

1. Introduction

Second order elliptic interface problems have many applications in engineering and science, in particular in fluid dynamics and material science [1–5]. Multiphase flow, crystal growth, modeling of the Stefan problem of solidification process, composite materials, cell and bubble deformation, and fiber suspension are few examples of the interface problems to be mentioned. Owing to discontinuity of coefficients of differential equations and singular source term, the solution and its derivative of interface problem are nonsmooth or even discontinuous across the interface. Hence usual numerical methods for partial differential equations do not work efficiently for these problems. Such kind of problems has attracted attention of scientists from both theoretical and computational approach over the past decades. Several high-order methods have been developed to deal with interface problem in terms of discontinuous coefficients and/or singular source terms. In [6] LeVeque and Li have investigated second order accurate finite difference method for one-dimensional interface problem. For one-dimensional problem, the standard finite element method can be used and if the interfaces lie on the grid point, second order accurate solution can be gained [7]. However by modifying basis function, Li obtained a second order accurate finite element method in the maximum norm in [7]. The finite difference and finite element method for one-dimensional problem are not much different. In [8], Shin and Jung used spectral collocation method and radial basis function method for one-dimensional interface problems and obtained very accurate solutions using the similar ideas given by Loubenets et al. in [9]. For two-dimensional elliptic interface problem, in [6], the immersed interface method (IIM) is developed for an elliptic interface problem with discontinuous coefficients and singular source term. By adding an extra node to the standard finite difference 5-point stencil, Huang and Li derived a second order accurate method in [10]; but the coefficient matrix is nonsymmetric and the method is stable for problems with piecewise constant coefficient. This problem is solved by a new version of IIM in [11]. In [12], Angelova and Vulkov presented second and fourth order finite difference scheme for elliptic problem with intersecting interface which has discontinuous variable coefficient and singular source term. However they assumed that interfaces are straight lines. In [13], Li et al. investigated finite element method based on Cartesian triangulation for interface problem involving discontinuous coefficients which is second order accurate in the maximum norm. Hanke and Loubenets proposed an immersed finite element method for elliptic interface problem with discontinuous coefficients and singular source term and present its second order convergence in norm in [14]. Kwak et al. have introduced an immersed finite element method based on the “broken” -nonconforming piecewise linear polynomials on interface triangular element having edge average as degrees of freedom in [15]. These linear polynomials are broken to match the homogeneous jump condition along the interface which is allowed to cut through the element. They have proved optimal order of convergence in the - and -norms. Latige et al. presented second order accurate finite volume method for interface problem with discontinuous coefficients in [16]. This paper is a generalization of one-dimensional work given in [8] to two-dimensional elliptic interface problems including curve boundaries.

In this paper, we first derive weak formulation of second order elliptic interface problem along with jump conditions. Then we give an algorithm to solve numerically these problems in the simple domain, namely, rectangle with straight line interface. We generalize the algorithm for domain with a curved boundary and interface by using a transformation so-called Gordon and Hall map. For a complicated domain, the spectral element collocation method is adopted by decomposing the domain into some nonoverlapping subdomains and then the spectral collocation method is used in each subdomain. Finally some numerical experiments are given to show efficiency and accuracy of our method.

2. Pseudo-Spectral Collocation Method

In this section we present an algorithm for solving an elliptic interface problem by the spectral collocation method. First we start with an elliptic interface problem with a discontinuous constant coefficient and a singular source term defined on a simple domain with line interface and then we extend it to an elliptic interface problem with any arbitrary domain and interface. We review some simple facts including function spaces, interpolation operators, and spectral matrices for the use of the spectral collocation method from now on. The standard notations and definitions are used for the weighted Sobolev spaces equipped with weighted inner product and corresponding weighted norms , , where is the Legendre weight function when and Chebyshev weight function when . Let be the space of all polynomials of degree less than or equal to and let be the Legendre Gauss Lobatto (LGL) or Chebyshev Gauss Lobatto (CGL) points on such that . For Legendre case, is the set of zeros of , where is the th Legendre polynomial and the corresponding quadrature weights are given by

For Chebyshev case, is the set of zeros of , where is the th Chebyshev polynomial and the corresponding quadrature weights are given by

For any continuous function on , let denote its Lagrange interpolation at collocation points ; that is,

Let be the Lagrange basis functions of degree such that

Then

The pseudo-spectral derivative of a continuous function is defined to be the exact derivative of the interpolant of ; that is,

Then the pseudo-spectral derivative matrix is

Let be the vector valued function containing nodal values of at ; that is, ; then the derivative vector valued function of is . If the interval is replaced by , we can use the following linear transformation:

to find Gauss-points and the quadrature weights

Introducing

we easily obtain the following spectral matrix

The two-dimensional LGL and CGL nodes and weights are defined as

Let be the space of polynomials of degree less than or equal to with respect to each variable and . The basis functions are also defined

We reorder the LGL and CGL points from bottom to top and then from left to right such that for . Then pseudo-spectral derivative matrix on 2-dimensional space is defined via the Kronecker tensor product; that is,

where is identity matrix of the same order as .

Let be a quadrilateral domain and let be an interface separating the domain into two subdomains and , such that . Here, is referred to as an interface. The boundary of is denoted by and , . Consider the following elliptic interface problem:

where and , with

and is a 2-dimensional delta function with the support along the interface .

In order to solve this problem via the interface problem, we need to first derive the jump conditions. It is obvious from the given equation that there exists one jump discontinuity by the Dirac -function. Using the standard finite element argument [6, 9], we have the following classical formulation:

along with the following interface jump conditions:

where denotes the outward normal vector on . The interface jump is defined as follows:

where and are the traces of and , respectively, on . The existence and uniqueness of solution of (17) is studied in [17].

Here we consider the pseudo-spectral method for the problem (17)-(18) with and as piecewise continuous constants. Suppose that is a piecewise continuous function and is continuous on . Suppose that is the pseudo-spectral approximation solution to problem (17)-(18). Then the approximation solution of the interface problem (17) can be expressed by

Although it is possible to use different polynomial order approximation on each subdomain, we use the same polynomial order approximation, for the sake of simplicity. Hence we have the following equations:

which can be written in matrix-vector form as

where and . Here

Let , where , , and denote by the values of at nodal points on the interface, boundaries, and interior of domain , respectively. By jump conditions and , we have and , respectively, which can be written in matrix-vector form as

where for on . The boundary conditions can be imposed as

Now, the boundary and jump conditions can be represented as

We also note that

Combining two systems, we have the following linear system:

where and . The resulting algebraic system can be efficiently solved by direct and iterative methods.

We now generalize the above algorithm for the case of discontinuous varying functions and . We still assume that the domain is rectangle and the interface is straight line as before. In this case, the first equation in (17) can be expanded as follows:

Hence the pseudo-spectral method for (29) reads:

which can be written in matrix-vector form as

with

and , where and are diagonal matrices of functions , and evaluated at nodal points , respectively. In this case, the boundary and interface conditions remain unchanged.

The algorithm proposed before is to solve interface elliptic equations defined in a quadrilateral domain with straight line interface. One may generalize the algorithm to solve interface problems defined in a domain with curved boundaries using Gordon and Hall transformation [18, 19]. Here we briefly introduce the Gordon and Hall transformation on a simply connected domain for reader’s convenience (see [18, 19] for more details.).

Let be a vector-valued function of two independent variables and over a domain in the -plane whose range is in . We assume that is a continuous one-to-one transformation which maps onto the domain such that . Then we would like to construct a one-to-one function which matches on the boundaries of , so-called the boundary interpolant of :

Following the ideas given in [18, 19], one may choose the following simple transfinite bilinear Lagrange interpolant of :

Then we have the transformed function defined on the rectangular domain and the elliptic equation defined on the curved domain is also transformed to an elliptic equation defined on the rectangular domain so that we can apply our proposed method to the transformed equation. The transformed equation can be represented as

whereas the coefficients , , and are functions in and . Applying the spectral collocation method to the above equations we have the following matrix form:

where , , , and are diagonal matrices of corresponding function values evaluated at nodal points , respectively. The jump and boundary conditions are similarly given as the previous . For complex geometries we employ pseudo-spectral element method. First, we decompose the domain into some nonoverlapping subdomains and then apply the spectral collocation method for each subdomain.

Finally, we give an example of the Gordon-Hall transformation for the domain given in Figure 2. The boundary interpolant is given by

and the explicit form of the transformation in (34) reduces to

3. Numerical Results

In this section, we first give an example defined on rectangle domain with straight line interface as in Figure 1. Afterward examples are defined on more complicated domain in which we have to use spectral element collocation method to solve them. Denote by the discrete solution of interface problem and by , its errors. We present their and discrete norm which is defined, respectively, as follows:

Figure 1: Interface domain.

Figure 2: A curved domain.

Example 1 (straight line interface). Consider

with the following exact solution:

where

and ’s can be determined by the boundary and jump conditions. We have

Tables 1 and 2 show the spectral convergence in - and -norm errors of Legendre and Chebyshev approximation for and , respectively.

Table 1: Error discretization for Example 1 with .

Legendre
Chebyshev

4

8

12

16

20

Table 2: Error discretization for Example 1 with .

Legendre
Chebyshev

4

8

12

16

20

As we pointed out in Section 2, it is possible to use different polynomial order approximation on each subdomain. To do this for Example 1, we divide into intervals, into intervals, and into intervals corresponding to LGL or CGL points. For sake of simplicity we take . The numerical result is given in Table 3 when . We have similar results for . When the interface point is close to the endpoint, different polynomial order can be used to get accurate results.

Table 3: Error discretization for Example 1 with and different polynomial order approximation.

Legendre
Chebyshev

4

8

12

16

20

In the following we present some numerical experiments in which the domain is and interface is the circle . To use spectral element collocation method, we decompose the domain into twelve nonoverlapping subdomains as in Figure 3. It should be noted that there is no jump in solution and its normal derivative for the interface which is different from ; that is,

where is dotted line in Figure 3.

Figure 3: Domain decomposition for the interface examples.

Example 2 (circular interface 1 [6]). Consider the elliptic equation of the form

In this example, we have only singular source term. The exact solution is given by

where .

The interface conditions are given by

Table 4 also shows the spectral convergence in - and -norm errors of Legendre and Chebyshev approximation. The exact solution, approximated one, and its error for the Legendre case of this example are plotted in Figure 4.

Table 4: Error discretization for Example 2.

Legendre
Chebyshev

4

8

12

16

20

24

(a)

(b)

(c)

Figure 4: Exact solution, approximation solution, and its error for .

Example 3 (circular interface 2 [6]). Consider the elliptic equation

where

The exact solution is given by

In this example, we have varying discontinuous coefficient and singular source term as well. Tables 4 and 5 show the numerical results for both Legendre and Chebyshev approximation for different values of and . We observe that the errors in - and -norm are decay exponentially regardless of discontinuous coefficients and singular source term. The exact solution, approximation, and its error for the Legendre case are plotted in Figures 5 and 6, for different values of and .

Table 5: Error discretization for and for Example 3.

Legendre
Chebyshev

4

8

12

16

20

24

(a)

(b)

(c)

Figure 5: Exact solution, approximation solution, and its error for and .

(a)

(b)

(c)

Figure 6: Exact solution, approximation solution, and its error for and .

Example 4 (circular interface 3 with jump condition [6]). Consider Laplace equation

The exact solution is given

In this example we impose a jump in solution itself and also a jump in the normal derivative of as external constraints. Table 6 also shows the spectral convergence for the Legendre and Chebyshev pseudo-spectral element method although it has jump in solution and its normal derivative Table 7. The exact solution, approximated one, and its error for the Legendre case of this example are plotted in Figure 7.

Table 6: Error discretization for and for Example 3.

Legendre
Chebyshev

4

8

12

16

20

24

Table 7: Error discretization for and for Example 4.

Legendre
Chebyshev

4

8

12

16

20

24

(a)

(b)

(c)

Figure 7: Exact solution, approximation solution, and its error for .

4. Concluding Remarks

In this paper, we proposed the pseudo-spectral collocation method for second order elliptic interface problems with discontinuous coefficients and singular source term. First we derived interface conditions in variable and its normal derivative and two equations defined on each subdomain decomposing the whole domain into two subdomains based on the interface. Then we obtain very simple algorithm applying the spectral collocation method to each equation. It is shown that the proposed method can easily applied to interface problem defined on curved domain by using Gordon and Hall transformation. This means that the method can be easily extended to spectral element collocation method to solve interface problems defined on a complicated domain. The numerical experiments also showed that the method has the spectral convergence with high accuracy. Furthermore, the method can be adopted to solve Stokes or Navier-Stokes equations with interface which can be studied in a coming paper.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The second author was supported by Kyungpook National University fund 2013 and the third author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF2013R1A2009687).

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