Journal of Applied Mathematics

Volume 2012 (2012), Article ID 859315, 11 pages

doi:10.1155/2012/859315

Research Article

Bernstein-Polynomials-Based Highly Accurate Methods for One-Dimensional Interface Problems

Jiankang Liu, Zhoushun Zheng, and Qinwu Xu

School of Mathematics and Statistics, Central South University, Changsha 410083, China

Received 10 June 2012; Accepted 11 August 2012

Academic Editor: Francisco J. Marcellán

Copyright © 2012 Jiankang Liu 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.

A new numerical method based on Bernstein polynomials expansion is proposed for solving one-dimensional elliptic interface problems. Both Galerkin formulation and collocation formulation are constructed to determine the expansion coefficients. In Galerkin formulation, the flux jump condition can be imposed by the weak formulation naturally. In collocation formulation, the results obtained by B-polynomials expansion are compared with that obtained by Lagrange basis expansion. Numerical experiments show that B-polynomials expansion is superior to Lagrange expansion in both condition number and accuracy. Both methods can yield high accuracy even with small value of N.

1. Introduction

In this paper, we consider the following two-point boundary value problem:

with boundary conditions

where is the Dirac delta function and is the dipole source term. The function is allowed to be discontinuous at . For simplicity here we assume that is smooth function. Due to the presence of singular source, the solution possesses the following jump relations [1]:

where the jump notation is defined as

This problem is referred to as the interface problem and is used in various applications of physics, engineering, and biological sciences, see [2–4] and the references therein.

For interface problems, since sharp interfaces or local jumps exist within the solution domain, any high-order method, such as the spectral method, suffers from the Gibbs phenomenon [5]. Here we evoke items we care most for solving interface problems. In 1993, the immersed interface method (IIM) was proposed for interface problems [1]. It is a two-order finite difference method based on Cartesian grids by incorporating the jump relations into difference schemes. The authors have constructed the IIM-based ADI finite difference scheme for 2D nonlinear convection diffusion interface problems [6]. In [7], a high-order method was developed for both discontinuous coefficients and singular source based on finite element method. To enhance the accuracy, the modified Hermite polynomials are used for the basic functions in each element. The matched interface and boundary method was proposed in [8] for elliptic interface problems with discontinuous coefficients and singular source. In [9], the coupling interface method was developed for elliptic interface problems. Recently, Shin and Jung [5] developed the spectral collocation method for one-dimensional interface problems (1.1) with , in which Lagrange basis functions are chosen as the trial functions.

In this work, we also consider this problem and take Bernstein polynomials basis as the trial functions. Bernstein polynomials are useful polynomials in computer-aided geometric design because of their excellent properties [10]. Recently there are some works that used Bernestein polynomials as basis for numerically solving differential equations [11, 12], integral equations [13, 14], and so on, but none of them is about interface problems. Our method is different from Shin and Jung’s in three aspects.

(i)The trial functions are chosen as Bernstein polynomials due to its nice properties. These polynomials defined on an interval form a complete basis over the interval. Each of these polynomials is positive and their sum is unity. (ii)The Galerkin formulation is constructed for this problem. Since the Bernstein polynomials are algebraic polynomials, the mass matrix and stiff matrices can be evaluated efficiently. (iii)The B-polynomial-based collocation formulation is given collocated with both equidistant points and spectral points, respectively. Unlike Lagrange basis functions, the B-polynomial differential matrix can be computed easily and the condition number of resulting linear system is smaller than that of Lagrange basis, which is shown in the numerical examples.

The rest of this paper is organized as follows. In Section 2, the Bernstein polynomials on interval and its derivatives are given. The Galerkin formulation and B-polynomial-based collocation formulation are shown in Section 3. Two numerical examples are given and analyzed in Section 4 before the conclusions are made in Section 5.

2. Bernstein Polynomials Basis

The general form of Bernstein polynomials of th degree on interval is defined as [12]

where the binomial coefficients are given by , with for and . These B-polynomials of degree form a complete basis over the interval . It is easy to show that any given polynomial of degree can be expressed in terms of linear combination of the B-polynomials basis functions. The B-polynomials can be generated by a recursive definition:

The derivatives of the th degree B-polynomials are combinations of B-polynomials of degree , which can be formulated as

where we set if or . In favor of these recursive relations, the differentiation matrix of Bernstein basis can be evaluated conveniently, while the computation of differentiation matrix of high-order Lagrange basis function may suffer from certain difficulties [15].

3. Numerical Methods

In this section, we give the numerical method for solving interface problem (1.1)–(1.3). It includes Galerkin formulation and collocation formulation. Without loss of generality, we assume that only one interface exists in and is piecewise constant in each subdomain. The multiple interface can be handled analogously. Thus the entire domain is divided by into two parts and .

Suppose that the solutions in and are and , respectively, then the problem (1.1)–(1.3) can be regarded as the following two smooth problems:

together with jump conditions (1.3), which make (3.1) closed.

In each subdomain , the solution can be approximated by

where is the degree of B-polynomials on each subdomain. According to the interpolation property of B-polynomials at two endpoints, we can easily get and .

3.1. Galerkin Formulations

The variational formulation of (3.1) reads

where is arbitrary and satisfies certain boundary conditions.

Plugging (3.2) into (3.3) and replacing with B-polynomials basis produce

with .

Observing that , and , , have zeros at the endpoints, (3.4) can be reformulated as

Let in (3.5) and in (3.6), we get

Adding (3.7) and (3.8) together, combined with jump condition , yields

Another jump condition implies

Equations (3.5), (3.6), (3.9), and (3.10) form a linear system with equations and unknowns:

3.2. Bernstein Collocation Methods

Substitution of (3.2) into (3.1) produces residuals

In each interval and , define the collocation points:

Here the points in can be equidistant points, Legendre-Gauss-Lobatto (L-G-L) points, or Chebyshev-Gauss-Lobatto (C-G-L) points.

Note that and in (3.2) are known. Collocation of (3.12) at points yields

From (3.2) we can easily get

Jump condition and imply

Equations (3.14) and (3.16) produce the linear systems of equations with unknowns

4. Numerical Experiments

In this section, we give two examples to verify the accuracy of proposed numerical method. The first example is the one in which and , while in the second example both and are not zeros. We compare our results (B-polynomials-based) with that of Shin and Jung’s (Lagrange-polynomials-based) [5]. In all cases, we take and the resulting linear systems are almost block diagonal and solved by BiCGStab algorithm.

Example 4.1. Consider the interface problem given in [5]

with the following exact solution:

where , , , and . The jump conditions read , . s can be determined by the boundary and jump conditions.

Both Galerkin formulation and collocation method are used to solve this problem numerically. The condition number (Cond) of the resulting linear system is given in each computation. Table 1 gives the convergence analysis of Galerkin formulation with different coefficients and , in which the and norms are shown. It can be seen that the error reduces rapidly as the order of the B-polynomials increases.

The convergence analysis of collocation formulation is shown in Tables 2–5, in which the B-polynomials basis and Lagrange basis are compared. In each table, we use two types of collocation points to compare the results. It shows that the results of spectral collocation points (L-G-L or C-G-L points) are more accurate than the equidistant collocation points. Comparing Table 2 with Table 3, we can conclude that the condition number of linear systems derived from B-polynomials is much smaller than that derived from Lagrange polynomials. And the error from the former is smaller than the latter. Similar analysis can be got by comparing Table 4 with Table 5.

Table 1: Convergence analysis of Example 4.1 by Galerkin formulation.

,
,

Cond

Cond

4

6

8

10

12

Table 2: Convergence analysis of Example 4.1 by Bernstein collocation method (, ).

Equidistant collocation points
L-G-L collocation points

Cond

Cond

4

6

8

10

12

Table 3: Convergence analysis of Example 4.1 by Lagrange collocation method (, ).

Equidistant collocation points
L-G-L collocation points

Cond

Cond

4

6

8

10

12

Table 4: Convergence analysis of Example 4.1 by Bernstein collocation method (, ).

Equidistant collocation points
C-G-L collocation points

Cond

Cond

4

6

8

10

12

Table 5: Convergence analysis of Example 4.1 by Lagrange collocation method (, ).

Equidistant collocation points
C-G-L collocation points

Cond

Cond

4

6

8

10

12

Example 4.2. In this example, the solution has nonzero jump across . The following interface problem is considered

The jump conditions are , . The exact solution is

where , , , , , and s can be determined by boundary and jump conditions.

The similar convergence analysis results can be obtained compared with Example 4.1. Since the nonzero jump just affects the right-hand side of the resulting linear systems, the condition numbers in Tables 6, 7, 8, 9, and 10 are unchanged compared with that in Tables 1–5, while the error in Tables 6–10 is much larger than that in Tables 1–5. The regularity of the solution can affect the accuracy of the numerical algorithm enormously.

Table 6: Convergence analysis of Example 4.2 by Galerkin formulation.

,
,

Cond

Cond

4

6

8

10

12

Table 7: Convergence analysis of Example 4.2 by Bernstein collocation method (, ).

N

Equidistant collocation points
L-G-L collocation points

Cond

Cond

4

6

8

10

12

Table 8: Convergence analysis of Example 4.2 by Lagrange collocation method (, ).

Equidistant collocation points
L-G-L collocation points

Cond

Cond

4

6

8

10

12

Table 9: Convergence analysis of Example 4.2 by Bernstein collocation method (, ).

Equidistant collocation points
C-G-L collocation points

Cond

Cond

4

6

8

10

12

Table 10: Convergence analysis of Example 4.2 by Lagrange collocation method (, ).

Equidistant collocation points
C-G-L collocation points

Cond

Cond

4

6

8

10

12

5. Conclusions

In this paper, a new numerical method based on B-polynomials expansion is proposed for solving one-dimensional interface problems. We give two methods to evaluate the expansion coefficients, the Galerkin formulation, and the collocation formulation. Both methods can yield highly accurate results with small number of B-polynomials. In collocation method, the Lagrange polynomials are used to compare with B-polynomials. It is shown by numerical examples that B-polynomials are superior to Lagrange polynomials in both condition number and accuracy, especially when collocated with equidistant points. In theoretical aspect, since the B-polynomials basis is equivalent to power basis or Lagrange basis under certain invertible transformations, theoretical analysis of the proposed method may be done similarly, which is a part of our future research plan. The method can be extended to problems with multiple interfaces easily.

Acknowledgments

The authors thank the supports from the Institute of Applied Software of Central South University. The authors also thank the anonymous reviewers and the editor for their valuable comments. This work was supported partially by National Natural Science Foundation of China (no. 51174236), National Basic Research Program of China (no. 2011CB606306), and Hunan Provincial Innovation Foundation For Postgraduate (no. CX2011B080).

References

R. J. LeVeque and Z. L. Li, “The immersed interface method for elliptic equations with discontinuous coefficients and singular sources,” SIAM Journal on Numerical Analysis, vol. 31, no. 4, pp. 1019–1044, 1994.

R. H. Goodman and R. Haberman, “Chaotic scattering and the n-bounce resonance in solitary-wave interactions,” Physical Review Letters, vol. 98, no. 10, Article ID 104103, 2007.

G. B. Jacobs and W.-S. Don, “A high-order WENO-Z finite difference based particle-source-in-cell method for computation of particle-laden flows with shocks,” Journal of Computational Physics, vol. 228, no. 5, pp. 1365–1379, 2009.

Y.-X. Li, “Tango waves in a bidomain model of fertilization calcium waves,” Physica D, vol. 186, no. 1-2, pp. 27–49, 2003.

B.-C. Shin and J.-H. Jung, “Spectral collocation and radial basis function methods for one-dimensional interface problems,” Applied Numerical Mathematics, vol. 61, no. 8, pp. 911–928, 2011.

J. Liu and Z. Zheng, “IIM-based ADI finite difference scheme for nonlinear convection-diffusion equations with interfaces,” Applied Mathematical Modelling. In press

A. Loubenets, T. Ali, and M. Hanke, “Highly accurate finite element method for one-dimensional elliptic interface problems,” Applied Numerical Mathematics, vol. 59, no. 1, pp. 119–134, 2009.

Y. C. Zhou, S. Zhao, M. Feig, and G. W. Wei, “High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources,” Journal of Computational Physics, vol. 213, no. 1, pp. 1–30, 2006.

I.-L. Chern and Y.-C. Shu, “A coupling interface method for elliptic interface problems,” Journal of Computational Physics, vol. 225, no. 2, pp. 2138–2174, 2007.

R. T. Farouki and V. T. Rajan, “On the numerical condition of polynomials in Bernstein form,” Computer Aided Geometric Design, vol. 4, no. 3, pp. 191–216, 1987.

M. I. Bhatti and P. Bracken, “Solutions of differential equations in a Bernstein polynomial basis,” Journal of Computational and Applied Mathematics, vol. 205, no. 1, pp. 272–280, 2007.

D. Bhatta, “Use of modified Bernstein polynomials to solve KdV-Burgers equation numerically,” Applied Mathematics and Computation, vol. 206, no. 1, pp. 457–464, 2008.

B. N. Mandal and S. Bhattacharya, “Numerical solution of some classes of integral equations using Bernstein polynomials,” Applied Mathematics and Computation, vol. 190, no. 2, pp. 1707–1716, 2007.

S. Bhattacharya and B. N. Mandal, “Use of Bernstein polynomials in numerical solutions of Volterra integral equations,” Applied Mathematical Sciences, vol. 2, no. 33–36, pp. 1773–1787, 2008.

B. Costa and W. S. Don, “On the computation of high order pseudospectral derivatives,” Applied Numerical Mathematics, vol. 33, no. 1, pp. 151–159, 2000.